Question

# Find each of the following limits. If the limit is not finite, indicate or for one- or two-sided limits, as appropriate. lim_{trightarrow0}frac{5t^2}{cos t-1}

Limits and continuity
Find each of the following limits. If the limit is not finite, indicate or for one- or two-sided limits, as appropriate.
$$\lim_{t\rightarrow0}\frac{5t^2}{\cos t-1}$$

2021-01-17
Consider the limit:
$$\lim_{t\rightarrow0}\frac{5t^2}{\cos t-1}$$
If we try to evaluate the limit by direct substitution we get 0/0.
So, evaluate using l’Hôpital’s Rule
$$\lim_{t\rightarrow0}\frac{5t^2}{\cos t-1}=5\lim_{t\rightarrow0}(\frac{\frac{d}{dt}(t^2)}{\frac{d}{dt}(\cos t-1)})$$
$$=5\lim_{t\rightarrow0}(\frac{2t}{-\sin t})$$
We het 0/0 form, so apply again L'Hospital Rule
$$5\lim_{t\rightarrow0}(\frac{\frac{d}{dt}(2t)}{\frac{d}{dt}(-\sin t)})=5\lim_{t\rightarrow0}(\frac{2}{-\cos t})$$
$$=5(\frac{2}{-\cos0})$$
$$=5(-\frac{2}{-1})$$
$$=-10$$