First slide

Continuity

Question

$Let f (x) = \{\begin{array}{cl} \frac{{(e^{x} - 1)}^{2 n}}{\sin^{n} (\frac{x}{a}) {(\log (1 + (\frac{x}{a})))}^{n}} & for x \neq 0 \\ 16^{n} & for x = 0 \end{array}$ and f is a continuous at $x = 0$ then the value of a is

Moderate

A

16
B

2
C

8
D

4

Solution

$\underset{x = 0}{L t} f (x) = \underset{x = 0}{L t} {(\frac{e^{x} - 1}{x})}^{2 n} \frac{{(\frac{x}{a})}^{n}}{\sin^{n} (\frac{x}{a})} \times \frac{{(\frac{x}{a})}^{n}}{{(\log (1 + (\frac{x}{a})))}^{n}} . a^{2 n} = a^{2 n} = f (0)$
$∴ a^{2 n} = 16^{n} \Rightarrow a = 4$

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