Question

Let $f$ be a continuous function on R. If $f\left(\frac{1}{4^n}\right)=\left(\sin e^n\right)e^{-n^2}+\frac{n^2}{n^2+1}$ then $f\left(0\right)$ is -

• A. not unique
• B. 1
• C. data sufficient to find $f\left(0\right)$
• D. data insufficient to find $f\left(0\right)$

Answer 1

Answer: (B), (C)

The correct options are
B 1
C data sufficient to find $f\left(0\right)$

$f\left(0\right)=\underset{n\to \infty }{lim}f\left(\frac{1}{4^n}\right)=\underset{n\to \infty }{lim}\left[\sin \right(e^n).e^{-n^2}+\frac{n^2}{1+n^2}]$
$=\underset{n\to \infty }{lim}[\frac{\sin \left(e^n\right)}{e^{n^2}}+\frac{1}{1+1/n^2}]=\underset{n\to \infty }{lim}[\frac{\sin \left(e^n\right)}{e^n}.\frac{1}{e^{n^2-n}}+\frac{1}{1+1/n^2}]$
$=[1.\frac{1}{\infty }+1]=1$