Search any question & find its solution
Question:
Answered & Verified by Expert
If $f(x)=(x+1)^{\text {cotx }}$ is continuous at $x=0$, then what is $f(0)$ equal to?
Options:
Solution:
1333 Upvotes
Verified Answer
The correct answer is:
e
For a function to be continuous at a point the limit should exist and should be equal to the value of the function at that point.
Here point is $\mathrm{x}=0$
and $\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0}(x+1)^{\cot x}$
$=\lim _{x \rightarrow 0}(1+x)^{\cot x}=\lim _{x \rightarrow 0}(1+x)^{\frac{1}{x} \cdot x \cot x}$
$=\lim _{x \rightarrow 0}(1+x)^{\frac{1}{x} \lim _{x \rightarrow 0} \frac{x}{\tan x}}=e^{1}=e$
Since limiting value of $f(x)=e$, when $x \rightarrow 0, f(0)$ should also be equal to e.
Here point is $\mathrm{x}=0$
and $\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0}(x+1)^{\cot x}$
$=\lim _{x \rightarrow 0}(1+x)^{\cot x}=\lim _{x \rightarrow 0}(1+x)^{\frac{1}{x} \cdot x \cot x}$
$=\lim _{x \rightarrow 0}(1+x)^{\frac{1}{x} \lim _{x \rightarrow 0} \frac{x}{\tan x}}=e^{1}=e$
Since limiting value of $f(x)=e$, when $x \rightarrow 0, f(0)$ should also be equal to e.
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.