Answer
$-\infty$
Work Step by Step
$\displaystyle \lim_{x\rightarrow-\infty}\frac{3x^{2}}{x+5}= \displaystyle \lim_{x\rightarrow-\infty}[\frac{3x^{2}\div x^{2}}{(x+5)\div x^{2}}$
$=\displaystyle \lim_{x\rightarrow-\infty}\frac{3}{\frac{1}{x}+\frac{5}{x^{2}}}$
... the terms $\displaystyle \frac{1}{x}$ and $\displaystyle \frac{5}{x^{2}}$ both approach 0 when $x\rightarrow\infty,$
The numerator is constant, and the denominator approaches zero,
so the limit does not exist. It is either $\infty$ or $-\infty.$
Observe the function $\displaystyle \frac{3x^{2}}{x+5}$.
It has a positive numerator whether x is negative or positive.
But the numerator is negative when $ x\rightarrow-\infty$.
So, the function has negative values when $ x\rightarrow-\infty$
Our answer is: $\displaystyle \lim_{x\rightarrow-\infty}\frac{3x^{2}}{x+5}=-\infty$
