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Problem of the Week #87 - November 25th, 2013

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Chris L T521

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Jan 26, 2012
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Thanks again to those who participated in last week's POTW! Here's this week's problem!

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Problem: Evaluate $\displaystyle \lim_{x\to 0}\frac{1}{x}\int_0^x \left(1-\tan(2t)\right)^{1/t}\,dt$.

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Hint:
Use L'Hôpital's rule.


Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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Chris L T521

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Jan 26, 2012
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This week's problem was correctly answered by MarkFL and Pranav. You can find Mark's solution below.

We are given to evaluate:

\(\displaystyle L=\lim_{x\to 0}\left(\frac{1}{x}\int_0^x \left(1-\tan(2t)\right)^{1/t}\,dt \right)\)

Since we have the indeterminate form \(\displaystyle \frac{0}{0}\), application of L'Hôpital's rule yields:

\(\displaystyle L=\lim_{x\to 0}\left(\left(1-\tan(2x)\right)^{1/x} \right)\)

Taking the natural log of both sides (and applying the rules of logs as they apply to limits and exponents), we obtain:

\(\displaystyle \ln(L)=\lim_{x\to 0}\left(\frac{\ln\left(1-\tan(2x) \right)}{x} \right)\)

Since we have the indeterminate form \(\displaystyle \frac{0}{0}\), application of L'Hôpital's rule yields:

\(\displaystyle \ln(L)=2\lim_{x\to 0}\left(\frac{\sec^2(2x)}{\tan(2x)-1} \right)=-2\)

Converting from logarithmic to exponential form, we find:

\(\displaystyle L=e^{-2}\)

Hence, we conclude:

\(\displaystyle \lim_{x\to 0}\left(\frac{1}{x}\int_0^x \left(1-\tan(2t)\right)^{1/t}\,dt \right)=\frac{1}{e^2}\)
 
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