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Problem 30 Medium 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 - \sin x}{x - \tan x} $

Answer

-1 / 2

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Video Transcript

Yes. The first problem here, we want to use the hotel's rules. So we're giving X minus Synnex divided by X minus candy snacks. So based on this, we see that when we plug in X equals zero, we get 0/0, which is an indeterminant form. So we want to take the uh derivative of the top and the bottom that will give us one minus cosine X over one minus second squared X. But unfortunately when we plug in zero here, we'll get 0/0 again. So we have to use local house rule again, this will give us a sign effect. So we can tell things are getting simpler. Um And then in the denominator we end up having negative to seek an X times 2nd X. Tangent tax. A little complicated in the bottom. That's okay because we can plug in zero. And when we do that we get zero. Um If we consider the fact that tangent X is the same as Synnex over Cosine X. We can simplify this and we'll get one over negative two seeking cube X. So now that we have this, we can plug in zero and we get one over a negative too, so it's going to be negative one half as our final answer.