(a) How is the number e defined? (b) Use a calculator to...

(a) How is the number $e$ defined? (b) Use a calculator to estimate the values of the limits $$\lim _{h \rightarrow 0} \frac{2.7^{4}-1}{h} \quad \text { and } \quad \lim _{h \rightarrow 0} \frac{2.8^{*}-1}{h}$$ correct to two decimal places. What can you conclude about the value of $e ?$

(a) How is the number $e$ defined?
(b) Use a calculator to estimate the values of the limits
$$\lim _{h \rightarrow 0} \frac{2.7^{4}-1}{h} \quad \text { and } \quad \lim _{h \rightarrow 0} \frac{2.8^{*}-1}{h}$$
correct to two decimal places. What can you conclude about the value of $e ?$

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Key Concepts

Limit Definition of e

The number e is defined as the limit of (1 + 1/n)^n as n approaches infinity. This definition captures the idea of continuous growth and serves as a cornerstone in calculus, particularly in contexts involving compound interest and natural exponential functions.

Exponential Function Derivative

For a positive number a, the expression lim??0 (a^(h) - 1)/h represents the derivative of the function a? evaluated at x = 0, which is equal to ln(a). Setting ln(a) = 1 identifies the unique number e, as the base for which the rate of change of the exponential function at zero is exactly one.

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