Answer
$-0.25$
Work Step by Step
Step 1. See table, use a series of negative $x$ values, we can calculate the function values to estimate the limit as $-0.25$.
Step 2. See graph, we can confirm the result graphically as $\lim_{x\to -\infty}f(x)=-0.25$.
Step 3. Algebraically, $\lim_{x\to -\infty}f(x)=\lim_{x\to -\infty}\frac{\sqrt {x^2+4x}}{4x+1}=\lim_{x\to -\infty}\frac{-\sqrt {1+4/x}}{4+1/x}=-\frac{1}{4}$.
