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Q.
if $ f\left(x\right) =\displaystyle\lim_{n\to\infty}\left( \frac{tan \pi x^{2}+\left(x+1\right)^{n} sin x}{x^{2}+\left(x+1\right)^{n}}\right)$, then
Continuity and Differentiability
Solution:
$f \left(0\right)=\frac{0+1\times0}{0+1}=0$
$\displaystyle \lim_{x \to 0^-}f (x)=$$\displaystyle \lim_{x \to 0^-}$$\displaystyle \lim_{x \to \infty}$$\frac{tan\,\pi x^{2}+\left(x+1\right)^{n}\,sin\,x}{x^{2}+\left(x+1\right)^{n}}$
$=\displaystyle \lim_{x \to 0^-}$$\frac{tan\,\pi x^{2}}{x^{2}}$ (If $x \rightarrow0^-, x+ 1 < 1$)
$=\pi\,\therefore LHL \ne f (0)$
$\therefore f(x)$ is not continuous at $x = 0$ hence not differentiable also.