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Algebra Arithmetic Functions I.S.I. and C.M.I. Entrance ISI Entrance Videos

Problem on Function | TOMATO BStat Objective 720

Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on function. You may use sequential hints to solve the problem.

Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Function.

Problem on Function (B.Stat Objective Question )


Consider the function f(x)=\(tan^{-1}(2tan(\frac{x}{2}))\), where \(\frac{-\pi}{2} \leq f(x) \leq \frac{\pi}{2}\) Then

  • \(\lim\limits_{x \to \pi-0}f(x)=\frac{\pi}{2}\), \(\lim\limits_{x \to \pi+0}f(x)=\frac{-\pi}{2}\)
  • \(\lim\limits_{x \to \pi}f(x)=\frac{\pi}{2}\)
  • \(\lim\limits_{x \to \pi-0}f(x)=\frac{-\pi}{2}\), \(\lim\limits_{x \to \pi+0}f(x)=\frac{\pi}{2}\)
  • \(\lim\limits_{x \to \pi}f(x)=\frac{-\pi}{2}\)

Key Concepts


Equation

Roots

Algebra

Check the Answer


But try the problem first…

Answer:\(\lim\limits_{x \to \pi}f(x)=\frac{\pi}{2}\)

Source
Suggested Reading

B.Stat Objective Problem 720

Challenges and Thrills of Pre-College Mathematics by University Press

Try with Hints


First hint

f(x)=\(tan^{-1}(2tan{\frac{x}{2}})\)

Second Hint

\(\lim\limits_{x \to \pi}f(x)\)

\(=\lim\limits_{x \to \pi}tan^{-1}(2tan{\frac{x}{2}})=\frac{\pi}{2}\)

\(\lim\limits_{x \to \pi-0}f(x)\)

\(=\lim\limits_{x \to \pi-0}tan^{-1}(2tan{\frac{x}{2}})=\frac{\pi}{2}\)

Final Step

\(\lim\limits_{x \to \pi+0}f(x)\)

\(=\lim\limits_{x \to \pi+0}tan^{-1}(2tan{\frac{x}{2}})=\frac{\pi}{2}\)

So \(\lim\limits_{x \to \pi}f(x)=\frac{\pi}{2}\)

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