#### Question

If \[f\left( x \right) = x^2 + \frac{x^2}{1 + x^2} + \frac{x^2}{\left( 1 + x^2 \right)} + . . . + \frac{x^2}{\left( 1 + x^2 \right)} + . . . . ,\]

then at x = 0, f (x)

##### Options

has no limit

is discontinuous

is continuous but not differentiable

is differentiable

#### Solution

(b) is discontinuous

\[\text{We have}, \]

\[f\left( x \right) = x^2 + \frac{x^2}{1 + x^2} + \frac{x^2}{\left( 1 + x^2 \right)} + . . . + \frac{x^2}{\left( 1 + x^2 \right)} + . . . . , \]

\[\text{When x = 0 then } x^2 = 0\]

\[ \text { and } \frac{x^2}{1 + x^2} = 0\]

\[ \therefore f\left( 0 \right) = 0 + 0 + 0 + 0 . . . . . . . \]

\[ \Rightarrow f\left( 0 \right) = 0\]

\[\text { When, x } \neq 0\]

\[\text{Then,} x^2 > 0\]

\[\text { and }1 + x^2 > x^2 \]

\[ \Rightarrow 0 < \frac{x^2}{1 + x^2} < 1\]

\[ \therefore \lim_{x \to 0} f\left( x \right) = \lim_{x \to 0} \left( x^2 + \frac{x^2}{1 + x^2} + \frac{x^2}{\left( 1 + x^2 \right)} + . . . + \frac{x^2}{\left( 1 + x^2 \right)} + . . . . , \right)\]

\[ = \lim_{x \to 0} \left[ x^2 \left( 1 + \frac{1}{1 + x^2} + \frac{1}{\left( 1 + x^2 \right)} + . . . + \frac{1}{\left( 1 + x^2 \right)} + . . . . , \right) \right]\]

\[ = \lim_{x \to 0} \left[ x^2 \left( \frac{1}{1 - \frac{1}{1 + x^2}} \right) \right] \left[ \text{Sum of infinite series where}, r = \frac{1}{1 + x^2} \right]\]

\[ = \lim_{x \to 0} \left[ x^2 \left( \frac{1 + x^2}{x^2} \right) \right]\]

\[ = \lim_{x \to 0} \left( 1 + x^2 \right)\]

\[ = 1\]

\[ \therefore \lim_{x \to 0} f\left( x \right) \neq f\left( 0 \right)\]

\[ \therefore f\left( x \right) \text { is discontinuous at } x = 0\]