Question

The function $f\left(x\right)=\{\begin{array}{cc}1-2x+3x^2-4x^3+....to\infty ,& x\ne -1\\ 1& ,x=-1\end{array}$ is

• A. continuous and derivable at $x=-1$
• B. neither continuous nor derivable at $x=-1$
• C. continuous but not derivable at $x=-1$
• D. none of these

Answer 1

The correct option is B neither continuous nor derivable at $x=-1$

For $x\ne -1$, we have
$f\left(x\right)=1-2x+3x^2-4x^3+...\infty =(1+x{)}^{-2}={\left(\frac{1}{1+x}\right)}^2$
Now, ${lim}_{x\to -1}f\left(x\right)={lim}_{x\to -1}\frac{1}{{(1+x)}^2}=\infty$ $\Rightarrow f\left(x\right)$is not continous at $x=-1$
So, $f\left(x\right)$ is not derivable at $x=-1$
Hence $f\left(x\right)$ is neither continuous nor derivable at $x=-1$