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
ANSWERED
asked 2021-01-16
Find each of the following limits. If the limit is not finite, indicate or for one- or two-sided limits, as appropriate.
\(\lim_{t\rightarrow0}\frac{5t^2}{\cos t-1}\)

Answers (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
\(5\lim_{t\rightarrow0}(\frac{\frac{d}{dt}(2t)}{\frac{d}{dt}(-\sin t)})=5\lim_{t\rightarrow0}(\frac{2}{-\cos t})\)
\(=5(\frac{2}{-\cos0})\)
\(=5(-\frac{2}{-1})\)
\(=-10\)
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