Limit with integral, case infinity * 0

Question

I attached an image with a limit, how can I solve that limit? If I use mean value theorem for integral it will be indeterminate form infinity * zero.

Please do not use pictures for critical portions of your post. Pictures may not be legible, cannot be searched and are not view-able to some, such as those who use screen readers. — 5xum, Apr 20 '21 at 07:15
https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference — 5xum, Apr 20 '21 at 07:15
Even the colours can be done in MathJax! Try inserting the following into your question: $$\int_{\color{cyan}{a}}^{\color{purple}{b}} f(x) dx = f(\color{red}{c})(\color{purple}{b} - \color{cyan}{a})$$ — Theo Bendit, Apr 20 '21 at 07:19
@theo in some countries tg is the shorthand for tangent instead of tan — Ninad Munshi, Apr 20 '21 at 07:21
OP substitute $u = \tan^{n+1}x$ then move the limit into the integral. — Ninad Munshi, Apr 20 '21 at 07:22

score 0 · Accepted Answer · answered Apr 20 '21 at 07:28

0

Use $u=\tan^{n+1}x \implies \frac{du}{1+u^{\frac{2}{n+1}}} = (n+1)\tan^nxdx$

$$\frac{n}{n+1}\int_0^1\frac{du}{1+u^{\frac{2}{n+1}}} \longrightarrow \int_0^1\frac{du}{1+1} = \frac{1}{2}$$

answered Apr 20 '21 at 07:28

Ninad Munshi

1 Answers1