Question

determine the end - behavior of the following transcendental function by evaluating appropriate limits. then provide a simple sketch of the associated graph, showing asymptotes if they exist. ( f(x)=-2^{x}). find the correct end - behavior of the given function. select the correct answer below and, if necessary, fill in the answer boxes to complete your choice. a. (lim_{x
ightarrowinfty}(-2^{x})=square) and (lim_{x
ightarrow-infty}(-2^{x})=square) b. neither (lim_{x
ightarrowinfty}(-2^{x})) nor (lim_{x
ightarrow-infty}(-2^{x})) exists and neither is (infty) nor (-infty)

Explanation:

Step1: Analyze $\lim_{x

ightarrow\infty}(- 2^{x})$
As $x
ightarrow\infty$, the function $y = 2^{x}$ is an exponential - growth function. Since we have a negative sign in front, $\lim_{x
ightarrow\infty}(-2^{x})=-\infty$.

Step2: Analyze $\lim_{x

ightarrow-\infty}(-2^{x})$
As $x
ightarrow-\infty$, we know that $y = 2^{x}=\frac{1}{2^{-x}}$. So, $\lim_{x
ightarrow-\infty}2^{x}=0$. Then $\lim_{x
ightarrow-\infty}(-2^{x}) = 0$.

Answer:

A. $\lim_{x
ightarrow\infty}(-2^{x})=-\infty$ and $\lim_{x
ightarrow-\infty}(-2^{x}) = 0$