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(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 $?

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01:34

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 1

Derivatives of Polynomials and Exponential Functions

Derivatives

Differentiation

Harvey Mudd College

Baylor University

Idaho State University

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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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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