Practicing Success
In order that the function f(x) = (x + 1)cot x is continuous at x = 0, f(0) must be defined as |
f(0) = 0 f(0) = e f(0) = 1/ e none of these |
f(0) = e |
For continuity actual value must be equal to limiting value $A=\lim\limits_{x \rightarrow 0}(x+1)^{\cot x}$ $\log A=\lim\limits_{x \rightarrow 0} \cot x \log (x+1)$ $=\lim\limits_{x \rightarrow 0} \frac{\log (x+1)}{\tan x}$ $\left[\frac{0}{0} \text { form }\right]$ $=\lim\limits_{x \rightarrow 0} \frac{\frac{1}{x+1}}{\sec ^2 x}=1$ (By L' Hospital Rule) $\log A=1 \Rightarrow A=e^1=e$ For f(0) must be defined as f(0) = e. Hence (2) is the correct answer. |