Answer
L'Hospital's rule applies,
the limit diverges to $-\infty$
Work Step by Step
By Theorem 10.2 (see section 10-3 ),
we can calculate this limit by ignoring all powers of $x$ except the highest in both the numerator and denominator.
$\displaystyle \lim_{x\rightarrow-\infty}\frac{10x^{2}+300x+1}{5x+2}$ = $\displaystyle \lim_{x\rightarrow-\infty}\frac{10x^{2}}{5x}$ = $\displaystyle \lim_{x\rightarrow-\infty}\frac{2x^{2}}{x}$
This limit has form $\displaystyle \frac{\infty}{\infty}$, L'Hospital's rule applies (*).
$= \displaystyle \lim_{x\rightarrow-\infty}\frac{[2x^{2}]^{\prime}}{[x]^{\prime} }= \displaystyle \lim_{x\rightarrow-\infty}\frac{2\cdot 2x}{1 }=$
diverges to $-\infty$
(*) We did not need to apply the LH rule here.
We could have just reduced the $x$ factors and arrived at the same answer.
