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

${\displaystyle\lim_{x\to5}f\left(x\right)}$

Evaluate the following limits and identify the horizontal asymptotes (if any) for the function $f\left(x\right)=\frac{5x}{25x+3}$:

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

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

Select the correct relationship between $\epsilon$ and $δ$ to prove ${\displaystyle\lim_{x\to-8}\left|5x\right|=40}$ using the $ε-δ$ definition of a limit.

The radius of a right cylinder having a height of $15\text{ cm}$ and a surface area of $U\text{ cm}^2$ is given as $r\left(U\right)=\frac15\left(\sqrt{225+\frac{5U}{\pi}}-15\right)$. Calculate ${\displaystyle\lim_{U\to0^{+}}}r\left(U\right)$ and provide an interpretation.

Use the following theorem to evaluate $\underset{x\to 0}{\lim }\frac{\sin 39x}{9x}$:

$\underset{x\to 0}{\lim }\frac{\sin x}{x}=1$

On the interval $(0, 15),$ locate the points where the function $f$ has discontinuities. For each discontinuity, indicate which continuity conditions are not met.

Let $g\left(x\right)=\begin{cases}\frac{x^3-1}{x-1} & \text{if }x\ne1\\ a & \text{if }x=1\end{cases}$

For what value of $a$ is $g\left(x\right)$ continuous at $x=1$?