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Problem 72 Hard Difficulty

Illustrate l'Hospital's Rule by graphing both $ f(x)/g(x) $ and $ f'(x)/g'(x) $ near $ x = 0 $ to see that these ratios have the same limit as $ x \to 0 $. Also, calculate the exact value of the limit.

$ f(x) = 2x \sin x $, $ g(x) = \sec x - 1 $

Answer

$=\lim _{x \rightarrow 0} \frac{2+2}{1}=4$

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Video Transcript

So let's take the two functions were given. That's alfa backs equals Um two x syntax and G fx equals second X. And it's one. Mhm. We don't care to look at these functions, we more care about their quotient. So F of X over G fx and F crime of acts over jeep crime of acts. And we want to look more closely at the point X equals zero and we see that they approached the same value and that value is for that makes sense too, because um if we look at these values here, um F prime of X over G prime of X is going to end up equaling um a trigger metric function. And then when we plug in zero we end up getting four. So that is the correct answer.