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I.S.I. and C.M.I. Entrance

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

Check the Answer


But try the problem first…

Answer: does not exist

Source
Suggested Reading

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\( h \to 0\) doesn‟t exist

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