First slide

Continuity

Question

The function $f (x) = \frac{{(3^{x} - 1)}^{2}}{sinx . \ln (1 + x)}, x \neq 0$ , is continuous at x = 0. Then the value of f(0) is

Moderate

A

$2 \log_{e} 3$
B

${(\log_{e} 3)}^{2}$
C

$\log_{e} 6$
D

none of these

Solution

$\begin{array}{l} Given f (x) is continuous at x = 0 . Therefore, \\ \lim_{x \to 0} f (x) = f (0) \\ or \lim_{x \to 0} \frac{{(3^{x} - 1)}^{2}}{sinxln (1 + x)} = f (0) \\ or f (0) = \lim_{x \to 0} \frac{\frac{{(\frac{3^{x} - 1}{x})}^{2}}{x}}{(\frac{sinx}{x}) (\frac{\ln (1 + x)}{x})} = {(\ln 3)}^{2} \end{array}$

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