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I don't have idea how to make this limit, i read it in a math contest. I think that is a limit that could be attacked by method of Riemann's sums. $\lim_{x\to 0} \int _0 ^ {x} (1- \tan (2t) ) ^ {\frac{1}{t}}\ dt$

Can you help me?

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    Just note for $|x|$ small, the integrand is positive and bounded above by 1; thus the integral $I$ satisfies $0\le I\le |x|$, when $|x|$ is small.2012-05-09

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Note that $|x|<\pi/8\implies 0\leq\left|\int_0^x (1-\tan(2t))^{1/t}dt\right|\leq \int_{-|x|}^{|x|} |1-\tan(2t)|^{1/t}\leq\int_{-|x|}^{|x|} 1dt=2|x|$ and since $2|x|\to 0$, the conclusion follows by the Squeeze theorem.

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    @DavidMitra Thanks for the catch. Dividing by two turned into multiplying in my head.2012-05-09
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First lets observe that

$\lim_{t \to 0} (1- \tan (2t) ) ^ {\frac{1}{t}}=e^{-2}$

Thus the function $f(t)=(1- \tan (2t) ) ^ {\frac{1}{t}}$can be extended continuously to $0$ by setting $f(t)=e^{-2}$.

Then by FTC

$\lim_{x \to 0}\dfrac{\int _0 ^ {x} (1- \tan (2t) ) ^ {\frac{1}{t}}\ dt}{x}= f(0)=e^{-2}$

which implies $\lim\limits_{x \to 0}\int _0 ^ {x} (1- \tan (2t) ) ^ {\frac{1}{t}}\ dt =0$