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Numerade Educator

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Problem 14 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{\tan 3x}{\sin 2x} $

Answer

$\frac{3}{2}$

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Video Transcript

So here we see that if we plug in zero into tangent of three of X over a sign of two of x, we just end up getting um We get 0/0. So instead we can do two options. We can not use low casserole. Um And to do that we would separate the limit or we could use low potatoes role. So using logic tells you would actually be simpler in this case. So let's do that. Um when you take the derivative of the numerator, What you end up getting is three times second squared reacts And then that's gonna be divided by two times could sign two X. Uh then we see that X is going to be equal to zero because we're evaluating the limited zero. So this right here When x equals zero is going to equal free and then this one X equals zero is going to end up giving us too. So we see that 3/2 will be our final answer and we could get this doing the other method. It would just be um, less quick. So whenever you can use the hotel's role, oftentimes it can be quicker, you just have to determine what would be best to use.