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  • If     <img alt="\mathrm{f(x)=x^2+\frac{x^2}{1+x^2}+\frac{x^2}{\left(1+x^2\right)^2}+\ldots+\frac{x^2}{\left(1+x^2\right)^n}+\ldots,~ then ~at ~x=0, f(x)}" src="https://entrancecorne

If     \mathrm{f(x)=x^2+\frac{x^2}{1+x^2}+\frac{x^2}{\left(1+x^2\right)^2}+\ldots+\frac{x^2}{\left(1+x^2\right)^n}+\ldots,~ then ~at ~x=0, f(x)}

Option: 1

has no limit


Option: 2

is discontinuous


Option: 3

is continuous but not differentiable


Option: 4

is differentiable.


Answers (1)

best_answer

Solution For  \mathrm{x \neq 0} , we have
\mathrm{ f(x)=x^2+\frac{\left(x^2 / 1+x^2\right)}{1-\left(1 / 1+x^2\right)}=x^2+1 } 
For \mathrm{ x=0, f(x)=0 }
Thus, \mathrm{ f(x)=\left\{\begin{array}{r}1+x^2, x \neq 0 \\ 0, x=0\end{array}\right. }

Clearly, \mathrm{ ~\lim _{x \rightarrow 0^{\infty}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)=1 \neq f(0)~ So,~ f(x) ~is ~discontinuous ~at ~x \neq 0 }

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manish

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