Answer
On the graph, we can see that both ratios have the same limit as $x \to 0$
$\lim\limits_{x \to 0}\frac{f(x)}{g(x)} = \frac{1}{4}$
Work Step by Step
On the graph, we can see that both ratios have the same limit as $x \to 0$
$\lim\limits_{x \to 0}\frac{f(x)}{g(x)} = \lim\limits_{x \to 0}\frac{e^x-1}{x^3+4x} = \frac{0}{0}$
We can apply L'Hospital's Rule.
$\lim\limits_{x \to 0}\frac{f'(x)}{g'(x)} = \lim\limits_{x \to 0}\frac{e^x}{3x^2+4} = \frac{1}{0+4} = \frac{1}{4}$
Therefore:
$\lim\limits_{x \to 0}\frac{f(x)}{g(x)} = \frac{1}{4}$
