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Numerade Educator

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Problem 15 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_{t\to 0} \frac{e^{2t} - 1}{\sin t} $

Answer

This limit has the form
$\frac{0}{0} \cdot \lim _{t \rightarrow 0} \frac{e^{2 t}-1}{\sin t} \stackrel{\Perp}{=} \lim _{t \rightarrow 0} \frac{2 e^{2 t}}{\cos t}=\frac{2(1)}{1}=2$

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Video Transcript

so we can either use the reptiles role or not use the hotel's role. Um In this case it's much easier to use low petals rule because we see once we plug in zero we get 0/0. And it's often trickier to deal with exponential functions and triggered a metric functions. So in this case we're just going to use um lovely calle's role. So we take the derivative of the numerator and we get to e. The two T. We take the derivative of the denominator. Um The derivative of sine T. Is co sign T. So we do that. Now this allows us to evaluate the function at T. equals zero. So when we do that this is going to end up giving us as a result. Um two E 20 which is one Provided by the co sign of zero which is so to eat of zero is going to be two times one rather. And then coastline of zero is just one that's going to end up giving us two divided by one. Which is to and that will be our final answer for love. Cause a real problem