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# Problem on Limit | ISI B.Stat Objective | TOMATO 728

Try this beautiful problem based on calculas from TOMATO 728 useful for ISI B.Stat Entrance. You may use sequential hints to solve the problem.

Try this beautiful problem based on Limit, useful for ISI B.Stat Entrance

## Problem on Limit | ISI B.Stat TOMATO 728

The limit lim $\int\frac {h}{(h^2 + x^2)}$dx (integration running from $x =-1$to $x = 1$) as$h \to 0$

• equals 0
• equals $\pi$
• equals $-\pi$
• deoes not exist

### Key Concepts

Limit

Calculas

trigonometry

But try the problem first…

Source

TOMATO, Problem 728

Challenges and Thrills in Pre College Mathematics

## Try with Hints

First hint

Now, $\int{h}{(h^2 + x^2)}$dx (integration running from $x = -1$ to $x = 1$)
Let, $x$ = h tany

$\Rightarrow dx = h sec^2y dy$
$\Rightarrow$ $x = -1$, $y = -tan^{-1}(1/h)$ and $x = 1$, $y = tan^{-1}(1/h)$

$\Rightarrow \int \frac{h}{(h^2 + x^2)}$dx =$\int \frac{h(hsec^2ydy)}{h^2sec^2y}$ (integration running from$y = -tan^{-1}(1/h)$ to $y = tan^-1(1/h))$

= y (upper limit =$tan-1(1/h)$) and lower limit = $-tan^-1(1/h)$

= $2tan^-1(1/h)$

Can you now finish the problem ……….

Second Hint

Now, lim $2tan^-(1/h)$ as$h \to 0$ doesnâ€Ÿt exist

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