A function is defined as $$fx=limn$$→∞ $$cos2nx$$, if x $$1which$$ of the following does not hold good?

Correct option is 1,2,3undefined

Questions

A function is defined as $f (x) = \lim_{n \to \infty} \{\begin{cases} \cos^{2 n} x, if x < 0 \\ \sqrt[n]{1 + x^{n}}, if 0 \leq x \leq 1 \\ \frac{1}{1 + x^{n}}, if x > 1 \end{cases}$ which of the following does not hold good?

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Continuous at x = 0 but discontinuous at x = 1

Continuous at x = 1 but discontinuous at x = 0

Continuous both at x = 1 and x = 0

Discontinuous both at x = 1 and x = 0

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detailed solution

Correct option is A

f0-=limn→∞ limn→0- cos2xn = a value lesser than 1∞=0f0+= limn→∞ limn→0+ 1+xn1n=1Also, f(0) = 1. So, the function is discontinuous at x = 0.Further, f1-=1; f1+=0; f1=1.So, the function is discontinuous at x = 1.

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A function is defined as fx=limn→∞ cos2nx, if x<01+xnn, if 0≤x≤111+xn, if x>1which of the following does not hold good?

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A function is defined as $f (x) = \lim_{n \to \infty} \{\begin{cases} \cos^{2 n} x, if x < 0 \\ \sqrt[n]{1 + x^{n}}, if 0 \leq x \leq 1 \\ \frac{1}{1 + x^{n}}, if x > 1 \end{cases}$ which of the following does not hold good?