First slide

Continuity

Question

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?

Moderate

A

Continuous at x = 0 but discontinuous at x = 1
B

Continuous at x = 1 but discontinuous at x = 0
C

Continuous both at x = 1 and x = 0
D

Discontinuous both at x = 1 and x = 0

Solution

$\begin{array}{l} f (0^{-}) = \lim_{n \to \infty} [\lim_{n \to 0^{-}} {(\cos^{2} x)}^{n}] \\ = {(a value lesser than 1)}^{\infty} = 0 \\ f (0^{+}) = \lim_{n \to \infty} [\lim_{n \to 0^{+}} {(1 + x^{n})}^{\frac{1}{n}}] = 1 \end{array}$
Also, f(0) = 1. So, the function is discontinuous at x = 0.
$Further, f (1^{-}) = 1; f (1^{+}) = 0; f (1) = 1 .$
So, the function is discontinuous at x = 1.

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