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In order that the function  \small f(x)=(x+1)^{\cos x} is continuous at x = 0, f(0) must be defined as

Option: 1

f(0)=0


Option: 2

f(0) = e 


Option: 3

f(0) = 1/ e 


Option: 4

none of these 
 


Answers (1)

best_answer

For continuity actual value must be equal to limiting value

\begin{aligned} & A=\lim _{x \rightarrow 0}(x+1)^{\cot x} \\ & \log A=\lim _{x \rightarrow 0} \cot x \log (x+1) \\ & =\lim _{x \rightarrow 0} \frac{\log (x+1)}{\tan x} \quad\left[\frac{0}{0} \text { form }\right] \end{aligned}

=\lim _{x \rightarrow 0} \frac{\frac{1}{x+1}}{\sec ^2 x}=1

(By L' Hospital Rule)

\log A=1 \Rightarrow A=e^1=e .

For  f(0) must be defined as f(0) = e . 


 

Posted by

avinash.dongre

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