Determine all vertical asymptotes $x=a$ of the function $f\left(x\right)=\frac{2x+3}{x^4-8x^3+16x^2}$. Evaluate $\underset{x\to a^{+}}{\lim }f\left(x\right)$, $\underset{x\to a_{}^{-}}{\lim }f\left(x\right)$, and $\underset{x\to a}{\lim }f\left(x\right)$ for each value of $a$.

$x=0$; ${\displaystyle\lim_{x\to0^{+}}}f\left(x\right)$ $=$ $\infty$; ${\displaystyle\lim_{x\to0^{-}}}f\left(x\right)$ $=$ $-\infty$; ${\displaystyle\lim_{x\to0}f(x)}$ $\text{does not exist}$

$x=4$; ${\displaystyle\lim_{x\to4^{+}}}f\left(x\right)$ $=$ $\infty$; ${\displaystyle\lim_{x\to4^{-}}}f\left(x\right)$ $=$ $-\infty$; $\displaystyle \lim_{x \to 4} f(x)$ $\text{does not exist}$

$x=0$; ${\displaystyle\lim_{x\to0^{+}}}f\left(x\right)$ $=$ $\infty$; ${\displaystyle\lim_{x\to0^{-}}}f\left(x\right)$ $=$ $\infty$; ${\displaystyle\lim_{x\to0}f(x)}$ $=\infty$

$x=4$; ${\displaystyle\lim_{x\to4^{+}}}f\left(x\right)$ $=$ $\infty$; ${\displaystyle\lim_{x\to4^{-}}}f\left(x\right)$ $=$ $\infty$; $\displaystyle \lim_{x \to 4} f(x)$ $=\infty$

$x=0$; ${\displaystyle\lim_{x\to0^{+}}}f\left(x\right)$ $=$ $\infty$; ${\displaystyle\lim_{x\to0^{-}}}f\left(x\right)$ $=$ $-\infty$; ${\displaystyle\lim_{x\to0}f(x)}$ $\text{does not exist}$

$x=4$; ${\displaystyle\lim_{x\to4^{+}}}f\left(x\right)$ $=$ $\infty$; ${\displaystyle\lim_{x\to4^{-}}}f\left(x\right)$ $=$ $-\infty$; $\displaystyle \lim_{x \to 4} f(x)$ $\text{does not exist}$