Refer to the graph of the function $f(x)$ to find the given limit if exists. If the limit does not exist, write "DNE".

$\underset{x\to 3}{\lim }f\left(x\right)$

Find the following limits and identify the horizontal asymptotes (if any) for the function $g\left(x\right)=\frac{4x^2+5x-3}{2x^2-7}$:

${\lim }_{x\to \infty }g\left(x\right)$

${\lim }_{x\to -\infty }g\left(x\right)$

Consider the transcendental function $f\left(x\right)=5e^{-x}$. What is the end behavior of this function as $x$ approaches $\infty$ and $-\infty$? Sketch a graph of the function, showing asymptotes if they exist.

Evaluate the limit as $x\to -\infty$ of the function $f\left(x\right)={\cot }^{-1}\left(5x\right)$ using its graph.

Select the correct relationship between $ϵ$ and $\delta$ to prove $\underset{x\to 0}{\lim }\left(5x^2\right)=0$ using the $\epsilon -\delta$ definition of a limit.