Let f(x)=ex−12sin⁡xalog⁡1+x4 for x≠0, and f(0) = 12.If f is continuous at x = 0, then the value of a is equal to

limx→0ex−12sin⁡xalog⁡1+x4=limx→0ex−12x⋅x2xa⋅sin⁡xaxa⋅log⁡1+x4x4⋅x4=4alimx→0f(x)=12 4a=12⇒a=3

Let f(x)=ex−12sin⁡xalog⁡1+x4 for x≠0, and f(0) = 12.If f is continuous at x = 0, then the value of a is equal to

limx→0ex−12sin⁡xalog⁡1+x4=limx→0ex−12x⋅x2xa⋅sin⁡xaxa⋅log⁡1+x4x4⋅x4=4alimx→0f(x)=12 4a=12⇒a=3

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Let $f (x) = \frac{{(e^{x} - 1)}^{2}}{\sin (\frac{x}{a}) \log (1 + \frac{x}{4})} for x \neq 0$ , and f(0) = 12.If f is continuous at x = 0, then the value of a is equal to

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Detailed Solution

$\lim_{x \to 0} \frac{{(e^{x} - 1)}^{2}}{\sin (\frac{x}{a}) \log (1 + \frac{x}{4})}$

$= \lim_{x \to 0} \frac{\frac{{(e^{x} - 1)}^{2}}{x} \cdot x^{2}}{\frac{x}{a} \cdot \frac{\sin (\frac{x}{a})}{(\frac{x}{a})} \cdot \frac{\log (1 + \frac{x}{4})}{\frac{x}{4}} \cdot \frac{x}{4}}$
=4a
$\lim_{x \to 0} f (x) = 12 4 a = 12 \Rightarrow a = 3$