I'm being unable to solve this exercise from the Demidovich's book where you need to evaluate limits involving definite integrals where the interval of integration is defined as a function of x.

How do I approach this kind of problems? Here I list one specific problem from Demidovich's book: $$\lim_{x\to+\infty} \frac{\int_{0}^{x} (\arctan (t))^2 dt}{\sqrt{x^2+1}}$$

Could you please demonstrate the approach on the problem above?


The limit in question is an indeterminate form (in this case $\frac\infty\infty$), so you would apply L'Hopital's rule. To differentiate the numerator, use the Fundamental Theorem of Calculus.

  • $\begingroup$ I applied L'Hospital's rule and I'm left with $\lim_{x\to+\infty} \frac{(\arctan(x))^2}{\frac{x}{\sqrt{x^2+1}}}$. How do I proceed? $\endgroup$ – Gogis May 4 '16 at 22:42
  • $\begingroup$ The denominator tends to $1$ because you can factor out $x$ from both top and bottom. As for the numerator, consult a plot of the arctan function to see that $\arctan x$ tends to $\pi/2$ as $x\to\infty$ $\endgroup$ – grand_chat May 4 '16 at 22:50
  • $\begingroup$ Yes, I see. Everything is clear to me now. Thanks. $\endgroup$ – Gogis May 4 '16 at 22:52

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