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Problem 16 Easy Difficulty

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0} \frac{x^2}{1 - \cos x} $

Answer

2

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Video Transcript

So once we directly substitute X equals zero. What we end up getting is 0/1 minus one. And that's clearly just 0/0, which is an intermediate form. So we are allowed to use local towels role when we take the derivative of the numerator, What we end up getting as a result is um a derivative X squared is going to be two X. And then the derivative of the denominator. Since we have one minus cosine X, we end up getting just Synnex. Now we can plug in zero. Although this is just going to give us 0/0. Again, another indeterminant form. But the beauty of locales role is that we can continue this. So now um using little zero, we take the derivative of this, we get to we take the derivative of the bottom, we get cosign X. Then based on this we can plug in zero. And that's going to end up giving us two divided by co sign of Zeros one. So our final answer is going to be too