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