Question

hw 2: problem 17
(1 point)
find the limit $lim_{x \to +\infty} \frac{e^{x} + e^{-x}}{e^{x} - e^{-x}}$
enter inf for $\infty$, -inf for $-\infty$, and dne if the limit does not exist.
answer:
hint: factor $e^{x}$ out of the numerator and denominator.

Explanation:

Step1: Factor $e^x$ from numerator

$\frac{e^x(1 + e^{-2x})}{e^x - e^{-x}}$

Step2: Factor $e^x$ from denominator

$\frac{e^x(1 + e^{-2x})}{e^x(1 - e^{-2x})}$

Step3: Cancel $e^x$ terms

$\frac{1 + e^{-2x}}{1 - e^{-2x}}$

Step4: Evaluate limit as $x\to+\infty$

As $x\to+\infty$, $e^{-2x}=\frac{1}{e^{2x}}\to0$. Substitute:
$\frac{1 + 0}{1 - 0}$

Answer:

1