Which one of the following function is discontinuous at $\mathrm{x}=1 ?$

Which one of the following function is discontinuous at $\mathrm{x}=1 ?$
  1. $f(x)=\sin ^2 x+\tan ^2 x+\cos ^2 x-\sec ^2 x$
  2. $f(x)=\frac{1}{1+2^{\sin x}}$
  3. $f(x)= \begin{cases}\frac{x-1}{|x-1|+2(x-1)^2}, & x \neq 1 \\ 1, & x=1\end{cases}$
  4. $f(x)=e^x+5$

Solution

$f(x)=\left\{\begin{array}{cc}\frac{x-1}{|x-1|+2(x-1)^2} & x \neq 1 \\ 1 & x=1\end{array}\right.$ $\Rightarrow f(x)=\left\{\begin{array}{cc}\frac{1}{2 x-1} & x>1 \\ \frac{1}{2 x-3} & x < 1 \\ 1 & x=1\end{array}\right.$ $\begin{aligned} & \lim _{x \rightarrow 1^{+}}+f(x)=\lim _{h \rightarrow 0} \frac{1}{2(1+h)-3}=\lim _{h \rightarrow 0} \frac{1}{2 h-1}=-1 \\ & \text { and } f(1)=1\end{aligned}$ $\because \quad \lim _{x \rightarrow 1^{+}} f(x) \neq f(1)$ $\therefore \mathrm{f}(\mathrm{x})$ is not continuous on $\mathrm{x}=1$.

Asked in: AP EAMCET 2023 (18 May Shift 1)

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