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Numerade Educator

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Problem 89 Hard Difficulty

If $ f' $ is continuous, use l'Hospital's Rule to show that
$$ \displaystyle \lim_{h\to 0} \frac{f(x + h) - f(x - h)}{2h} = f'(x) $$
Explain the meaning of this equation with the aid of a diagram.

Answer

$a=\frac{4}{3}$, $b=-2$

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Video Transcript

Mhm. For this problem. Um Since the limit is in the indeterminate form 0/0, we'll use low Patel's rule and that's going to give us um F of X. Class H minus F of X minus H. All over to H. Um And since we're applying the limit as H goes to zero, what we end up getting is when we take the derivative, since we can't do that, we're going to get F prime have expressed H I'm 1 uh-. That's gonna be, we're gonna multiply that by negative one. Eventually it'll be plus after prime of X minus H. And this is all gonna be over to. So based on this, when we plug in H equals zero. Now we get two F prime of X over two. So it's just going to give us f prime of X. Um as our final answer and therefore we've proven this.