First slide

Continuity

Question

$If f (x) = \{\begin{cases} \frac{xlog cosx}{\log (1 + x^{2})}, x \neq 0 \\ 0, x = 0 \end{cases}$ , then

Moderate

A

f(x) is not continuous at x = 0
B

f(x) is continuous at x = 0
C

f(x) is continuous at x = 0 but not differentiable at x = 0
D

f(x) is differentiable at x = 0

Solution

$\begin{array}{l} We have \lim_{x \to 0} \frac{f (x) - f (0)}{x - 0} = \lim_{x \to 0} \frac{\log cosx}{\log (1 + x^{2})} \\ = \lim_{x \to 0} \frac{\log (1 - 1 + cosx)}{\log (1 + x^{2})} \frac{1 - cosx}{1 - cosx} \\ = \lim_{x \to 0} \frac{\log \{1 - (1 - cosx)\}}{1 - cosx} \frac{1 - cosx}{\log (1 + x^{2})} \\ = \lim_{x \to 0} \frac{\log [1 - (1 - cosx)]}{1 (1 - cosx)} \frac{2 \sin^{2} \frac{x}{2}}{4 (\frac{x}{2})} \frac{x^{2}}{\log (1 + x^{2})} = - \frac{1}{2} \\ Hence, f (x) is differetiable x = 0 . \end{array}$
Hence, (2') and (4) are the correct answers.

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