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

${\displaystyle\lim_{x\to3}f\left(x\right)}$

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

$\lim_{x\rightarrow\infty}g\left(x\right)$

$\lim_{x\rightarrow-\infty}g\left(x\right)$

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