Question

evaluate the following limits.
(a) $lim_{x
ightarrowinfty}\frac{4}{e^{x}+4}=square$
(b) $lim_{x
ightarrow-infty}\frac{4}{e^{x}+4}=square$
note: if needed, enter oo for $infty$ and -oo for $-infty$.
question help: video

Explanation:

Step1: Analyze limit as $x\to\infty$

As $x\to\infty$, $e^{x}\to\infty$. So we have $\lim_{x\to\infty}\frac{4}{e^{x}+4}$.
Since the denominator $e^{x}+4\to\infty$ and the numerator is a constant 4, we know that $\lim_{x\to\infty}\frac{4}{e^{x}+4}=0$.

Step2: Analyze limit as $x\to-\infty$

As $x\to-\infty$, $e^{x}\to0$. So we consider $\lim_{x\to - \infty}\frac{4}{e^{x}+4}$.
Substitute $e^{x}\to0$ into the expression, we get $\lim_{x\to - \infty}\frac{4}{e^{x}+4}=\frac{4}{0 + 4}=1$.

Answer:

(a) 0
(b) 1