Question

If $f\left(x\right)=\underset{n\to \infty }{lim}\frac{\tan \pi x^2+(x+1{)}^n\sin x}{x^2+(x+1{)}^n},$ then at $x=0,$ which of the following is/are NOT TRUE?

• A. $f$ has removable (missing point) discontinuity
• B. $f$ has removable (isolated point) discontinuity
• C. $f$ has non-removable (finite type) discontinuity
• D. $f$ is continuous

Answer 1

Answer: (A), (B), (D)

$f$ is continuous

In the immediate neighbourhood of $x=0,$ $f\left(x\right)$ is defined as
$f\left(x\right)=\{\begin{array}{cc}\sin x& \text{for}x>0\\ 0& \text{for}x=0\\ \frac{\tan \pi x^2}{x^2}& \text{for}x<0\end{array}$

Now, $\underset{x\to 0^{+}}{lim}\sin x=0$
and $\underset{x\to 0^{-}}{lim}\left(\frac{\pi \tan \pi x^2}{\pi x^2}\right)=\pi$
LHL $\ne$ RHL
Hence, it's non-removable (finite type) discontinuity.