This exercise explores the difference between the limit

Chapter 4, Problem 84E

(choose chapter or problem)

$\lim _{x \rightarrow \infty}\left(1+\frac{1}{x^{2}}\right)^{x}$

and the limit

$\lim _{x+\infty}\left(1+\frac{1}{x}\right)^{x}=e$

a. Use l'Hôpital's Rule to show that

$\lim _{x+\infty}\left(1+\frac{1}{x}\right)^{x}=e$

b. Graph

$f(x)=\left(1+\frac{1}{x^{2}}\right)^{x} \text { and } g(x)=\left(1+\frac{1}{x}\right)^{x}$

together for $x \geq 0$.How does the behavior of $f$ compare with that of $g$? Estimate the value of $\lim _{x \rightarrow \infty} f(x)$.

c. Confirm your estimate of $\lim _{x \rightarrow \infty} f(x)$ by calculating it with l'Hôpital's Rule.

Equation Transcription:

$\lim\limits_{{x} \rightarrow {\infty{}}}(1\ +\frac{1}{{x}^{2}}{)}^{x}$

$\lim\limits_{{x} \rightarrow {\infty{}}}(1+\frac{1}{x}{)}^{x}=e$

$f(x)=(1+\frac{1}{{x}^{2}}{)}^{x}\ and\ g(x)\ =(1+\frac{1}{x}{)}^{x}$

$x\geq{}0$

$f$

$g$

${lim}_{x\rightarrow{}\infty{}}f(x)$

Text Transcription:

lim over x rightarrow infinity (1 + 1/x^2)^x

lim over x rightarrow infinity (1 + 1/x^2)^x = e

f(x) = (1 + 1x2)^x and g(x) =(1+1x)^x

x geq 0

f

g

lim_x rightarrow infinity f(x)

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