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Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity2 Marks
MCQ
For the function $f (x)=\left\{\begin{array}{ll}\frac{ e ^{1 / x}-1}{ e ^{1 / x}+1}, & , x \neq 0 \\ 1 & , x=0\end{array}\right.$, which of the following is correct?
  • A
    $\lim _{x \rightarrow 0^{+}} f(x)=-1$
  • B
    $f (x)$ is continuous at $x=0$
  • C
    $\lim _{x \rightarrow 0^{-}} f(x)=1$
  • ✓
    $f (x)$ is not continuous at $x=0$
✓

Answer

Correct option: D.
$f (x)$ is not continuous at $x=0$
(D)
$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} \frac{e^{\frac{-1}{h}}-1}{e^{\frac{-1}{h}}+1}=\lim _{h \rightarrow 0} \frac{e^{\frac{1}{{ }^{\frac{1}{h}}}-1}}{\frac{1}{e^{\frac{1}{h}}}+1}=\frac{0-1}{0+1}=-1$
$\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0} \frac{e^{\frac{1}{h}}-1}{e^{\frac{1}{h}}+1}=\lim _{h \rightarrow 0} \frac{1-\frac{1}{e^{\frac{1}{h}}}}{1+\frac{1}{e^{\frac{1}{h}}}}=\frac{1-0}{1+0}=1$
$\therefore \quad \lim _{x \rightarrow 0^{-}} f (x) \neq \lim _{x \rightarrow 0^{+}} f (x)$
$\therefore f (x)$ is not continuous at $x=0$.

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