Question

Let $f\left(x\right)=\underset{n\to \infty }{lim}\frac{(x^2+2x+3+\sin \pi x{)}^n-1}{(x^2+2x+3+\sin \pi x{)}^n+1}$. Then

• A. $f\left(x\right)$ is continuous and differentiable for all x $\in$ R
• B. $f\left(x\right)$ is continuous but not differentiable for all x $\in$ R
• C. $f\left(x\right)$ is discontinuous at infinite number of points
• D. $f\left(x\right)$ is discontinuous at finite number of points

Answer 1

The correct option is A $f\left(x\right)$ is continuous and differentiable for all x $\in$ R

$f\left(x\right)=\underset{n\to \infty }{lim}\frac{{(x^2+2x+3+\sin \pi x)}^n+1-2}{{(x^2+2x+3+\sin \pi x)}^n+1}$

$=\underset{n\to \infty }{lim}(1-\frac{2}{{(x^2+2x+3+\sin \pi x)}^n+1})$

$x^2+2x+3={(x+1)}^2+2$

$\Rightarrow x^2+2x+3\ge 2$

$-1\le \sin \pi x\le 1$

$\Rightarrow (x^2+2x+3+\sin \pi x)>1$

$\Rightarrow \underset{n\to \infty }{lim}(1-\frac{2}{{(x^2+2x+3+\sin \pi x)}^n+1})$

$=1-\frac{2}{\infty }=1-0=1$

$\Rightarrow f\left(x\right)=1=$constant function

$\Rightarrow$continuous on R

$f^{\prime }\left(x\right)=0=$constant function

$\Rightarrow f$ is differentiable on R