Lamassu were protectors of not only kings and palaces, but of every single human being. Mythical Guardians that Influenced Christianity Akkadians associated Lamassu with the god Papsukkal (the messenger god), and the god Isum (a fire-god, herald of the Babylonian gods) with Shedu. The cults of Lamassu and Shedu were very common in households from the Sumerian to Babylonian period, and they became associated with many royal protectors in different cults. In the Epic of Gilgamesh, they were considered as protective because they encompass all life within them. No matter if they are in a female or male form, Lamassu always represent the parent-stars, constellations, or the zodiac. In the Epic of Gilgamesh and Enuma Elis, both Lamassu and Aspasu (Inara) are symbols of the starry heavens, constellations, and the zodiac. University of Chicago Oriental Institute. Human-headed winged bull, otherwise known as a Shedu, from Khorsabad. She corresponds with the Greek goddess Artemis. Lamassu, as a celestial being, is also identified with Inara, the Hittite-Hurrian goddess of wild animals of the steppe and the daughter of the Storm-god Teshub. The ancient origins of the legendary griffin.Sometimes a Lamassu is portrayed as a female deity, but usually it is presented with a more masculine head. Other names for Lamassu are Lumasi, Alad, and Shedu. The first recorded Lamassu comes from circa 3,000 BC. Lamassu frequently appear in Mesopotamian art and mythology. Lamassu in the Akkadian language means “protective spirits.” Celestial Beings The Mesopotamians believed that Lamassu frightened away the forces of chaos and brought peace to their homes. Additionally, they were the guardians who inspired armies to protect their cities. At the same time, another winged creature was made to keep watch at the throne room entrance. Other statues of the mythical beasts belong to cities like ancient Dur Sharrukin (current Khorsabad, Iraq).Įvery important city wanted to have Lamassu protect the gateway to their citadel. The winged beasts from Nimrud in Iraq (the ancient city of Kalhu) also became very famous when Lamassu there were damaged in 2015. The most famous colossal statues of Lamassu have been excavated at the sites of the Assyrian capitals established by King Assurnasirpal II (reigned 883 – 859 BC) and King Sargon II (reigned 721 – 705 BC). They were believed to be very powerful creatures, and served both as a clear reminder of the king’s ultimate authority and as symbols of protection for all people. Lamassu are human-headed, eagle-winged, bulls or lions that once protected cities in Mesopotamia.
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That being said, Kandy reiterated a point that Gottmik made which is that this final four are very close with each other and that she met lifelong friends from the show. The main point of Kandy’s story is her childhood in the Bronx and having to fight for so much in her life and how that explains her drag persona. – Next, Kandy came to the stage to explain to Ru that even she sometimes can’t understand what she’s saying. Her parents also sent a good luck message, to which Gottmik explained that they’ve always been supportive of her journey.Ĩ:21 p.m. Gottmik explained that Paris was actually one of the first celebrity clients she had as a makeup artist. As a treat, Paris sent a good luck video message to Gottmik congratulating her and wishing her luck for the win. First up was Gottmik, who won Snatch Game impersonating Paris Hilton. Ru sat down with each of the finalists to check in on their journey through the season. – After the catwalks, Ru came to the stage of the Ace Hotel to introduce the evening to the outdoor audience with the reigning queen Jaida Essence Hall as correspondent from the Drag Race Drive-In. And in the third and final walk, they wore their premiere grand finale look: Gottmik had an elaborate navy gown, Kandy came out in a gown as well with a huge peacock feather, Rosé went classic with a baby green and white Victorian look, and Symone closed in white and gold.Ĩ:08 p.m. They walked a second time in all red: Gottmik went with a red Keith Harrington look, Kandy came out in a red leather mini dress with gold chains, Rosé went gothic with a red branch wig, and Symone finished in a flirty red fringe dress made of nails. – Opening the night was Ru in gold tulle performing her new song “New Friends Silver, Old Friends Gold.” Afterward, the final four were introduced in their best black & white looks: Gottmik was in all white makeup with a black gown, Kandy went in a white bodysuit with a train and an Elvira-esque wig, Rosé went in a trippy Alice in the Wonderland take on the classic Audrey Hepburn look, and Symone arrived in an enormous afro wig and an oversized bandana gown. But it was stunning not to see Shangela in the final lip sync - and hard not to think about how differently things may have played out had her gamesmanship not gone up in dragonfire.Below, check out our minute-by-minute “RuPaul’s Drag Race” recap of episode 15 to find out who won the lip syncs for the crown in “Grand Finale.” After following along with our live updating blog, be sure to sound off in the comments section with your reactions to VH1’s reality TV show and then discuss whether the right queen joined the “RuPaul’s Drag Race” winners list.Ĩ:01 p.m. No one can say Trixie's win over Kennedy Davenport wasn't deserved - she more than made up for a lackluster season 7 by showing off her legit musical talents and comedy skills ("Snatch Game" not included), not to mention that iconic "Red for Filth" look. While All Stars 3 continued the tradition of "lip syncing for your legacy," the queens were blindsided two times over - once when BenDeLaCreme chose to send herself home after starting the season with four challenge wins in a row, and then again when the eliminated queens returned as a jury to narrow the final four queens down to the top two. As it turns out, Khaleesi Laquifa Wadley may have forseen her fate. Martin, because the third turn of All Stars had more dramatic twists and turns than Shangela's favorite HBO epic-turned-perpetual metaphor. Some of these are references to the first game, and Orange Company remains a running joke. So Jowy spends some time coming up with names. I mean, of course, let's definitely let fifteen year olds lead a company of soldiers. You just have to do about 80 points of damage to get Flik to relent. This one's easy to parse, it's a Wild Attack. : Okay, maybe I should show you what I've really got. Some of Suikoden V's duels get close to it, though, but that's because that game's Duel system is much more fleshed out. To extrapolate that, that duel is easily the hardest in the series, even if you prepare for it. This duel is super easy, and this initial taunt is an Attack, so let's Wild Attack back at him.Īctually, for that matter, ALL Duels in this game are piss-fucking-easy, and have absolutely nothing on the Pahn-Teo duel. They've changed absolutely nothing in-between games aside from cinematic flair and updated graphics. Azure, if you can even just wound me, I'll consider you a warrior, okay? : I don't want anyone slowing us down, so I'll give you a test. Ignoring this little brat, let's go talk to Viktor. Whether they want to be involved or not Azure and Jowy are a part of this war now. If you answer the first option, Tsai simply knocks Azure and Jowy out and we continue on with the plot. You and I would not have a chance of standing up against him. Luca Blight possesses strength that is not of this world. Luca: Finished? My sword still thirsts for blood!! You! Burn this place down! Burn this whole village to the ground!! Luca: Is that right? In that case, act like a pig.Įasily the most iconic action in the game alone, when Luca kills a woman after he makes her degrade herself to plead for her life. Luca: All you worms can do is plead! You really want to live so badly? This is one of the most iconic scenes in Suikoden II, mostly because of the dialogue. Upon our return to Ryube Village, we find it has been completely burned to the ground and ransacked by Luca Blight's army. Tsai comes stock with a spell scroll of Clay Guardian, the first Earth spell, and it actually works in this game. Like I said before, you sometimes get a choice of where to place battle characters either in your convoy or in your party. He comes with a Unicorn Rune attached, but it's not fixed. He gets great stats but has low HP to compensate it. Tsai is a pretty good, but kind of average fighter. : The Fire Spears, right? I heard that Luca Blight attacked Toto Village. Past where the bird's nest was is a bridge, which takes us on the path towards Tsai's place. Apple tells us he's living near Ryube Village, perhaps in the forest. He says there's a man who lives near here called Tsai of the Divine Spear.Īlways need more money. Viktor tells us the current stock of Fire Spears they have are rusted over, and that we need to go get more. Yup, the Fire Spears are back as a plot point although, they're not nearly as important or effective as they were in the first game. : I'm going to look for a good place to lay some traps. : I'm going to Muse to request reinforcements. The Unicorn Brigade massacre, Captain Rowd's betrayal, Toto Village burning down. But I always thought someday I could go back to Highland. GOAT JRPGs, still salty that the series is no more while inferior pretenders (naming no names) are churned out annually.: I thought I was a Highlander and then they hunted us down. And this is only two-thirds or so through the game.įuck, this thread is making me want to screw my backlog and do yet another Suikoden series playthrough. The catharsis when you finally take this motherfucker down is incredible, and would make a worthy climax to any JRPG. The buildup, the music, the three-party boss fight, the genius moment with the fireflies (my headcanon is that the wooden amulet used for that part of the plan was the same one Pilika sent Riou and Jowy to buy just before Luca sacked her just seems poetic that way) and then the final duel. The whole night ambush is one of my favourite RPG setpieces of all time. He's simply a bloodthirsty, incredibly dangerous human who is only taken out using the combined efforts of pretty much your entire army. He isn't a True Rune bearer (who are usually the strongest fighters in the Suikoden series) and heck, I don't think he even uses runes at all (apart from maaaaybe a bog-standard Fire Rune in his sword). He's by far the most memorable boss I've ever fought in a JRPG, and he managed to achieve it without multiple forms, apocalyptic special attacks, or any of the usual cliches. Vietnamese Restaurants for Families in City Center East.Dinner Restaurants in Bella Vista / Queen Village.Bella Vista / Queen Village Restaurants.Asian Restaurants in Bella Vista / Queen Village.Restaurants with Outdoor Seating in Philadelphia.Restaurants for Special Occasions in Philadelphia.Restaurants for Group Dining in Philadelphia.Kid Friendly Restaurants in Philadelphia.Food Delivery Restaurants in Philadelphia.
The main al-Qa`ida forums have adopted this trend and are active on Twitter, promoting their official Twitter accounts on the main jihadist web forum pages. The most prominent visual element is the use and role of the specific black banner crafted by the Islamic State of Iraq, which today is the most credible identity marker for pro-al-Qa`ida jihadist groups. Elements shown in jihadist videos are thus most appealing to initiated consumers who can read and identify the greater ideology at work. This trend reflects to a great extent a specific zeitgeist, a contemporary as well as generational shift from texts and scripts to a “visual literacy.” Ideology is presented by iconographic, habitual, and rhetorical means. These jihadists decided to engage in physical violence after being active within the virtual dominions of al-Qa`ida where exposure to the media had an impact on their personal lives and understanding of religious conduct. Recent martyr biographies reveal that the growth in social media use has led to a new generation of jihadists. Twitter: The New Beacon for Jihadist Activity Online Through the mining of this data, this article identifies a content sharing network of more than 20,000 active Twitter accounts and a collection of YouTube video files that have been viewed nearly 450,000 times. Data for this article is derived from analysis based on more than 76,000 tweets, containing more than 34,000 links to web-based content. Using an interdisciplinary approach to the analysis of jihadist propaganda, this article demonstrates how jihadist groups are using Twitter to disseminate links to video content shot on the battlefield in Syria and posted for mass consumption on YouTube. This article discusses the emergence of jihadist social media strategies, explains how the Syrian jihadist group Jabhat al-Nusra (JN) has used Twitter to disseminate content, and analyzes content shared by JN. Indeed, within the complex network of interconnecting sites, Twitter has become the main hub for the active dissemination of links guiding users to digital content hosted on a range of websites, social media platforms, and discussion forums. The importance given to Twitter by jihadist groups, however, was highlighted in the 11th issue of Inspire magazine, which was released by al-Qa`ida in the Arabian Peninsula (AQAP) in May 2013. The classical jihadist discussion forums remain the vital hub for authoritative and cohesive propaganda online. The role of the “media mujahidin” has been approved, sanctioned and encouraged with the release of suggested strategies, although not all jihadists have perceived the move away from the traditional discussion forum websites as positive. As the range of online platforms has expanded, jihadist groups have increasingly used sites such as Twitter, Facebook, YouTube, and Tumblr. Online media platforms facilitate a blend of audiovisual media interspersed with writings that further sanction and explain specific ideological dimensions of jihadist activity. Other big-name video game franchises celebrating anniversaries this year include Pokemon and Resident Evil, both of which have now been around for 20 years.Almost all forms of online media allow, enable and empower users to generate their own content and interact by posting comments, questions or responses. Crush 40 is scheduled to perform at the 25th anniversary party. Interestingly, Crush 40 lead singer Johnny Gioeli wrote on Facebook last year that Sega may be working on a Sonic "anniversary" game. Speaking at a SXSW panel in March, a Sega representative wouldn't say if a new game would be announced at the event, but said fans should "totally come to this party," as Sega will make a "very special announcement." As announced previously, attendees will get an "exclusive first look at the future of Sonic." However, whether this means a new game reveal or something else remains to be seen. If Sega is planning to officially announce a new Sonic game, the reveal could happen at the company's 25th anniversary party in San Diego at Comic-Con this summer. The hint you're looking for is elsewhere. Great eye, but those squares in the Sonic post are just for design. Diabetes: Neuropathy, a complication of diabetes, can cause muscle weakness leading to the development of flat feet.Inappropriate Footwear: Prolonged use of high heels or shoes without proper arch support can lead to flat feet over time.Arthritis: Inflammatory conditions like rheumatoid arthritis can contribute to the development of flat feet.Ĭertain causes of flat feet may come as a surprise as they aren't commonly associated with this condition:.Injury: Damage to the muscles, tendons, or bones in the feet can cause flat feet.Pregnancy: Hormonal changes during pregnancy can loosen ligaments and result in flat feet.Obesity: Increased body weight can put excessive pressure on the arches and lead to their collapse.Aging: As individuals age, muscles and tendons can lose their properties, leading to fallen arches.Alternatively, it can develop over time due to a variety of reasons such as: Each case differs depending on the patient’s condition and severity.įor more information on flat feet or how to get relief with this condition, please contact our office today.Flat feet can be a congenital condition, meaning individuals are born with it. An orthopedic surgeon may create arches in your foot, repair tendons, or fuse bones and joints together to gain function back to your feet. In severe cases, surgery may be recommended, but it is often used as a last resort. Lifestyle changes such as diet and exercise to manage weight and reduce the amount of pressure on your feet.Wearing appropriate footwear with adequate arch support.This will help to strengthen ligaments, tendons, and muscles of the foot and ankle Physical therapy can help when foot pain is due to an injury.Performing stretches to keep the muscles and tendons flexible.Braces and support may also be fitted to you Custom arch supports, or orthotics, to take the pressure off your arches and provide cushioning support.Medication to reduce swelling and inflammation and alleviate pain.However, if you do experience pain, your podiatrist will recommend treatment for you to reduce pain and discomfort. TreatmentĪs mentioned above, if you aren’t experiencing any pain or other symptoms, treatment is often not required. Other tests, such as a gait test, may also be done to see the extent of the condition and evaluate other contributing factors of the patient’s flat foot. X-rays and other imaging tests may be performed to evaluate the extent of the deformity and the condition of the tissues, muscles, and bone of the foot. For those who seek relief, scheduling an appointment with a podiatrist can provide you with a proper diagnosis and treatment plan to relieve you of pain and discomfort.Ī podiatrist will conduct a clinical examination and request a detailed medical history. If you aren’t experiencing any pain or other symptoms, treatment is often not required. Pain along arches after standing or walking for extended periods.Associated deformities such as bunions or hammertoes. Pain or discomfort in the foot, leg, or lower back.The most common symptom is pain felt in the arch, heel, foot, and ankle. However, this is not the case for all patients. SymptomsĪ large number of people with flat feet may not experience any pain or symptoms and will not need treatment. Other causes can include heredity, tight tendons, arthritis, diabetes, excess weight, overuse, strain, and age. Injury and trauma can also cause abnormalities in the tendons, ligaments, and muscles of the foot and ankle, resulting in a flat foot. People with severely overpronated feet are predisposed to developing a rigid flat foot. Flat feet are common in young children, but as their body grows, the arch and the muscles supporting it strengthen and will become a normal arch by adulthood. Many patients who have a flat foot were born with a flexible flat foot that didn’t fully develop with age. A rigid flat foot means that the arch is flat while supporting weight and also when relaxed. A flexible flat foot means the arches give way when the foot supports the body but regains an arch once the foot is relaxed. There are two general types of flat feet, flexible and rigid. This is known as pronation, and the degree of pronation varies from patient to patient. Most people have a degree of arch support in the foot that acts like a shock absorber while walking or standing. The lack of an arch characterizes a flat foot, meaning the arches have collapsed, and the entire sole touches the floor while standing or walking. It can occur on one foot or both and develop from a variety of causes. Flat feet, also referred to as overpronated feet or fallen arches is a common postural deformity with various degrees of physical impact. But wait, what is that monstrosity on your back? All your hard work is undone by that 15-year-old backpack that has certainly seen better days. Be careful of patent or overly shiny shoes, you don’t want to be going for that night club toilet attendant look.īy now you should be a well dressed man-about-town, catching the ladies eyes and oozing confidence as you go about your business. Not much to be said here, black is suitable for any occasion, browns can look great, especially with a more vintage style suit, but you’ll be struggling when you get invited to that swanky black tie event. Chunky soles guarantee that schoolboy look. Leave that trusty Casio at home, and save the bright orange G-shock for the gym.Īnd by slim, I mean rounded or pointed toes with a low profile sole. Stick to metal watches in a silver or grey, with matching metal bracelets, or a dark leather strap, avoiding digital watches. Simple, understated design is an excellent choice, adding to the outfit rather than overpowering it. A must have accessory, a classic timepiece can be a functional and stylish way to finish off an outfit. Darker, more traditional colours (black, charcoal, navy) are once again the most versatile.īarring rings, watches are one of the few pieces of jewellery a man can get away with. Choose something that compliments your body shape, spend the time to take your measurements properly. Once again, size and fit are what counts here. Never buy short sleeves, long sleeves rolled up will always be a better option. Try to get as perfect a fit as possible, ensuring the collar isn’t too tight or loose, in an ideal world a tailor made shirt would be the preferred choice, however great results can be achieved by picking the correct size. Make sure the quality of the shirt is up to scratch, thin polyester types tend to look see-through and will end up resembling a school shirt. Perfect for wearing with a multitude of suits, the white shirt can also complement nearly anything you wear. Darker shades have the versatility of being able to stand up to evening wear, whilst not giving the appearance of being overdressed during the day. Too baggy and you’ll look like a kid wearing their fathers clothes, too skinny and you’ll end up looking like One Direction. Stay away from ripped ‘fashion’ jeans, those adorned with decals and logos and think about fit. The trusty workhorse of the wardrobe, it is worth investing in a quality pair of jeans you can rely on for years to come. A simple pair of jeans will compliment a massive variety of clothing and can be reused in countless different outfits. They are quite simply engrained in the human brain as a fashionable, practical, no-nonsense, all-purpose garment. There’s a reason blue jeans have never disappeared. A simple navy blazer goes with pretty much anything. Sticking with classic dark tones allows for much greater coordination with other clothes you may already have. Throw on top of a jeans and t-shirt for an effortless smart casual look, or team with a pair of suit trousers to up the ante for those ‘no jeans’ type affairs. Here is a quick run-down of 10 timeless pieces every gentleman should think about adding to their wardrobe.Ī good, well-fitting jacket is the cornerstone of any stylish mans collection. First impressions count in life, whether it be impressing a date, making a good impression at interview or even just encounters with new people. Pre-compiled packages at conan.io package manager: OpensSSL for Windows, Linux, OSX, Android Versions for Solaris 2.5 - 11 SPARC and X86 Reproducible builds with latest MinGW-w64, 64/32-bit, static/dynamic libs and executable. Pre-compiled Win32/64 libraries without external dependencies to the Microsoft Visual Studio Runtime DLLs, except for the system provided msvcrt.dll. Comes in form of self-install executables. Works with MSVC++, Builder 3/4/5, and MinGW. Originally developed for the cryptology playground "CrypTool-Online". The Wasm execution happens using WebWorkers if the browser supports them. Uses Emscripten and xterm.js to emulate a terminal in your browser. OpenSSL 3.0 ported to WebAssembly (in October 2021). Third Party OpenSSL Related Binary Distributions Use these OpenSSL derived products at your own risk these products have not been evaluated or tested by the OpenSSL project. Alternatively consider becoming a corporate sponsor. To make a donation to the project directly go to our github sponsors page. In particular any donations or payments to any of these organizations will not be known to, seen by, or in any way benefit the OpenSSL project. The listing of these third party products does not imply any endorsement by the OpenSSL project, and these organizations are not affiliated in any way with OpenSSL other than by the reference to their independent web sites here. This service is primarily for operating systems where there are no pre-compiled OpenSSL packages. If you are such a user, we ask you to get in touch with your distributor first. Those are already well-known among the users of said distributions, and will therefore not be mentioned here. Note: many Linux distributions come with pre-compiled OpenSSL packages. The condition to get a link here is that the link is stable and can provide continued support for OpenSSL for a while. Some people have offered to provide OpenSSL binary distributions for selected operating systems. Help for Low Credit Scores +More All Help for Low Credit Scores Best Credit Cards for Bad Credit Best Personal Loans for Bad Credit Best Debt Consolidation Loans for Bad Credit Personal Loans if You Don't Have Credit Best Credit Cards for Building Credit Personal Loans for 580 Credit Score Lower Personal Loans for 670 Credit Score or Lower Best Mortgages for Bad Credit Best Hardship Loans How to Boost Your Credit Score Taxes +More All Taxes Best Tax Software Best Tax Software for Small Businesses Tax Refunds Small Business +More All Small Business Best Small Business Savings Accounts Best Small Business Checking Accounts Best Credit Cards for Small Business Best Small Business Loans Best Tax Software for Small Business Personal Finance +More All Personal Finance Best Budgeting Apps Best Expense Tracker Apps Best Money Transfer Apps Best Resale Apps and Sites Buy Now Pay Later (BNPL) Apps Best Debt Relief Best Mortgages for Average Credit Score.Best Loans to Refinance Credit Card Debt.More details on software and accessibility are available at. 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Fair Isaac is not a credit repair organization as defined under federal or state law, including the Credit Repair Organizations Act. Many factors affect your FICO Scores and the interest rates you may receive. Equifax Credit Report is a trademark of Equifax, Inc. Learn moreįICO, myFICO, Score Watch, The score lenders use, and The Score That Matters are trademarks or registered trademarks of Fair Isaac Corporation. Your lender or insurer may use a different FICO ® Score than the versions you receive from myFICO, or another type of credit score altogether. All rights reserved.Īll FICO ® Score products made available on include a FICO ® Score 8, and may include additional FICO ® Score versions. To make the relationship better PayPal will remove the option to chose so there is no 1% FTF from Mastercard and they get to make a slight profit in the conversion with fees at the same time. Sneaky but effective.Ĭopyright ©2001- Fair Isaac Corporation. When Synchrony Bank offers no FTF on the PayPal 2% Cashback they still get hit with a 1% FTF from MasterCard and have to eat it. This is because PayPal and Synchrony Bank are in cahoots. There are threads where people reported not being able to opt-out of the conversion being done by PP when the PP MasterCard is used even though the PP MasterCard has no fee (I'll try to find the links). I know American Express doesn't allow DCC anyway, but as far as I know Discover does so it's not clear (to me) how that would work if the billing was long as your payment is one-time you should be fine in any case.Įdited to add: Beware of using the PP MasterCard for overseas Paypal transactions. The only cards I have with a link to this screen are MC and Visa it's missing for Discover and Amex. Last payment I made via Paypal for an international purchase IIRC during checkout I had the option to select where the converstion took place, but there's a separate setting ( really hard to find) if your payment will be recurring (note the 'classic' interface): If we make the denominator smaller, the value of the fraction gets larger: = 500, = 500000. divide 5 by 0.1, we get that it equals 50. For example, if we look at the fraction, i.e. We look for asymptotes at points where the denominator is zero because: when the denominator gets close to zero and becomes very small, it makes the value of the function very large. If a function has values on both sides of an asymptote, then it cannot be connected, so it must have a discontinuity at the asymptote. There are three types of discontinuities: Discontinuity 1: Asymptotic DiscontinuitiesĪsymptotes occur when a function approaches infinity at a specific value of x or y. Now, what would a discontinuous function look like? A function essentially is discontinuous when it has any “gap”. The most important is to recognize a continuous function when you see it. If you are confused by that, ignore it! You don’t need to learn all at once. There is a precise mathematical definition of continuity that uses limits. Intuitively, this definition says that small changes in the input of the function result in small changes in the output. Here’s an example of what a continuous function looks like: The basic idea of continuity is very simple, and the “formal” definition uses limits.īasically, we say a function is continuous when you can graph it without lifting your pencil from the paper. Limits and continuity are often covered in the same chapter of textbooks. The value of the function at the specific point we care about is not defined, like 0/0 (which is complete junk), or useless, like zero or infinite. In calculus, the most useful limits are like this one. What does the function approach when x approaches 1? It also approaches 1, right? It doesn’t matter that the function is other than 1 at that point! So, This function is the same as the one we saw before, but in this case it has a “hole” at x=1. Well, the point is that sometimes we don’t care what the function is at x=1.Īs an example of this, let’s consider the following function:ĭon’t let this notation intimidate you! This only means that this function equals x 2 when x is anything other than 1, and equals 0 when x equals 1. Why would you need to know what the function is approaching? You already know the function equals 1 when x equals 1, right? This is read “the limit as x approaches 1 of x squared equals 1”. We can see from the graph that when x approaches 1, the function f(x) approaches 1. Let’s turn to the graph of a function whose expression we know: I do this because we don’t necessarily know the value of function f at x=a. In this graph I drawed a big pink hole at the point (a,L). When x approaches the value “a” in the x axis, the function f(x) approaches “L” in the y axis. To start getting used to this idea, let’s turn to this graph: The idea behind limits is to analyze what the function is “approaching” when x “approaches” a specific value. For example, the function y = x 2 + 2 assigns the value y = 3 to x = 1, y = 6to x = 2, and y = 11 to x = 3. Using this function, we can generate a set of ordered pairs of ( x, y) including (1, 3),(2, 6), and (3, 11). We all know about functions, A function is a rule that assigns to each element xfrom a set known as the “ domain” a single element yfrom a set known as the “ range“. With an understanding of the concepts of limits and continuity, you are ready for calculus. Using limits, we’ll learn a better and far more precise way of defining continuity as well. For the math that we are doing in precalculus and calculus, a conceptual definition of continuity like this one is probably sufficient, but for higher math, a more technical definition is needed. One easy way to test for the continuity of a function is to see whether the graph of a function can be traced with a pen without lifting the pen from the paper. A function can either be continuous or discontinuous. ContinuityĬontinuity is another far-reaching concept in calculus. ” Symbolically, this is written f ( x) = 6. For example, given the function f ( x) = 3 x, you could say, “The limit of f ( x) as xapproaches 2 is 6. LimitĪ limit is a number that a function approaches as the independent variable of the function approaches a given value. We still use the Leibniz notation of dy/dx for most purposes. These two gentlemen are the founding fathers of Calculus and they did most of their work in 1600s. Gottfried Leibnitz is a famous German philosopher and mathematician and he was a contemporary of Isaac Newton. The concept of the Limits and Continuity is one of the most crucial things to understand in order to prepare for calculus. |
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