Let f be a function defined on R by f(x)=limn→∞ log⁡(3+x)−x2nsin⁡x1+x2n then

# Letbe a function defined on R by $f\left(x\right)=\underset{n\to \mathrm{\infty }}{lim} \frac{\mathrm{log}\left(3+x\right)-{x}^{2n}\mathrm{sin}x}{1+{x}^{2n}}$ then

1. A

$f$ is continuous on R

2. B

$f$ is continuous on

3. C

$f$ is continuous on

4. D

none of these

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### Solution:

If $|x|<1$ then  as  and if $|x|>1$

. So

$\underset{x\to 1+}{lim} f\left(x\right)=-\mathrm{sin}1,\underset{x\to 1-}{lim} f\left(x\right)=\mathrm{log}4$

$\underset{x\to -1+}{lim} f\left(x\right)=\mathrm{log}2,\underset{x\to 1-}{lim} f\left(x\right)=\mathrm{sin}1$

so $f$is continuous on  $\mathbf{R}~\left\{-1,1\right\}$

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