Question

If a function $f(x)$ is given by $f(x)=\frac{x}{1+x}+\frac{x}{(x+1)(2 x+1)}+\frac{x}{(2 x+1)(3 x+1)}+\ldots \infty$ then at $x =0$, $f(x)$

• A. has no limit
• B. is not continuous
• C. is continuous but not differentiable
• D. is differentiable

Answer 1

The Correct Option is B

Let $f(x)=\frac{x}{1+x}+\frac{x}{(x+1)(2 x+1)}+\frac{x}{(2 x+1)(3 x+1)}+\ldots \infty$
$=\displaystyle\lim _{n \rightarrow \infty} \displaystyle\sum_{r=1}^{n} \frac{x}{[(r-1) x+1](r x+1)}$
$=\displaystyle\lim _{x \rightarrow \infty} \displaystyle\sum_{r=1}^{n}\left[\frac{x}{[(r-1) x+1]}-\frac{1}{r x+1}\right]$
$=\displaystyle\lim _{n \rightarrow \infty}\left[1-\frac{1}{n x+1}\right]=1$
For $x =0$, we have $f ( x )=0$
Thus we have
$f(x) =
\begin{cases}
1, & x \neq 0\\
0, & x = 0
\end{cases}$
Clearly, $\displaystyle\lim _{x \rightarrow 0^{-}} f(x)$
$=\displaystyle\lim _{x \rightarrow 0^{+}} f(x) \neq f(0)$
So, $f (x)$ is not continuous at $x =0$