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  4. Let f be a continuous function on R such that f(1 / 4 n)=( sin en) e-n2+(n2/n2+1) Then the value of f(0) is

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Q. Let $f$ be a continuous function on $R$ such that $f(1 / 4 n)=\left(\sin e^{n}\right) e^{-n^{2}}+\frac{n^{2}}{n^{2}+1}$ Then the value of $f(0)$ is

Continuity and Differentiability

Solution:

As $f$ is continuous,
$f(0) =\displaystyle\lim _{x \rightarrow 0} f(x)$
$=\displaystyle\lim _{n \rightarrow \infty} f(1 / 4 n)$
$=\displaystyle\lim _{n \rightarrow \infty}\left(\left(\sin e^{n}\right) e^{-n^{2}}+\frac{1}{1+1 / n^{2}}\right)$
$=0+1=1$