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Problem 1 Easy Difficulty

(a) How is the number $ e $ defined?
(b) Use a calculator to estimate the values of the limits

$ \displaystyle \lim_{h\to 0}\frac {2.7^h - 1}{h} $ and $ \displaystyle \lim_{h\to 0}\frac {2.8^h - 1}{h} $

correct to two decimal places. What can you conclude about the value of $ e $?


a. $\lim _{h \rightarrow 0} \frac{e^{h}-1}{h}=1$
b. since $0.99<1<1.03,2.7<e<2.8$

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Video Transcript

hands Claire's when you read here. So he has a number such that the limit. Those beach approaches. Zero. You have a TSH minus one. Oh, her h This vehicle to one for part B. We have the limit using a calculator. Each approaches infinity 2.7 to the H power minus one over age is equivalent to a 0.99 The limit as each approaches infinity 2.8 to the H power minus one or her age is equivalent two 1.3 And you know that E is a number such that the slope of e of X is one at X equals zero and we also know the formula limit. Each approaches infinity. Many of beach minus one for her h is equal to the derivative. So he sought that the slope of A of X for a is equal to 2.7. It's less than one, and for a is equal to 2.8 is more than one. So it has to be between, um, 2.7 92.8 since the slope of A to X power when a equals E is one

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