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Numerade Educator



Problem 54 Hard Difficulty

Show that $ \displaystyle \lim_{t \to a} r(t) = b $ if and only if for every $ \varepsilon > 0 $ there is a number $ \delta > 0 $ such that
if $ 0 < \mid t - a \mid < \delta $ then $ \mid r(t) - b \mid < \varepsilon $


$\lim _{t \rightarrow a} r(t)=b$


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Video Transcript

First, we prove that if the limit T is able to be then for a every Gibson critics and zero, there is a number that is greater than zero are sad. If there is smaller than humans, it's absolutely of T minus a smaller than data. Then absolute value of our human speed is smarter than you're soon. The first oh any Gibson is greater than zero things limit. Rt is likely to be behalf limit. Axity is equal to be one limit. Why two? It's equal to B two. They made thank you is equal to B three. Here are t is equal to excuse me Why Chief 80 and be is equal to be one Me too three then by the definition of the limiter of function. But now from these three limits for any it soon from is the quarter it's gonna over three, right? By the definition of the limits were half there exists that one. That too got three such that yes, zero is more than she was. A It's more than 3 to 1 XT once B one is smaller than the product proud And if zero smaller than T minus a smaller than two white tea minus B to a small son. Absolutely product. The zero smaller than humanness. A smaller than two or three C g. When the speed three is smaller than excellent, then with lot Dad, it's difficult to you. Minimum value of Yeah. 12 yeah, three. Then, if zero is smaller than T minus, pay more than the other to half. Artie might be absolute value. This is equal to negative. 60 minus B one square us Y T minus B two square, us C T minus B three square So this is it's more in the it's a three hip strewn from square. This is equally true. You too. If some square over three, this is smaller than yes, sir. Then we approve. If, for every Epsilon greeted and zero, there is a data such that if zero is smaller than humanness, a smaller than Delta, then our team in space more than it's more than the limit R T is able to be. Yeah, for any tips greater than zero Vegas this data, and if zero is monitored, she might stay small smaller than data. We have rt oneness being smaller instrument than behalf 60 minus B one is smaller than rt minus b So this is more than six months for this same data and same Epson. Why two? I want to speak Chu. Yeah, smother small 13 minus b three. Yes, Mother. It's more then by the definition of the limits of our function we have limit. XT is equal to be one limit Y t is equal to be true. Limit is equal to three So the limit Archie is equal to limit angsty limit Y t limit Say cheese. This is equal to you. Want to be three? This is equal to