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Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity2 Marks
MCQ
If $f(x)=\left\{\begin{array}{c}\frac{x}{e^{\frac{1}{x}}+1}, \text { when } x \neq 0 \\ 0, \text { when } x=0\end{array}\right.$, then
  • A
    $\lim _{x \rightarrow 0^{+}} f(x)=1$
  • B
    $\lim _{x \rightarrow 0^{-}} f(x)=1$
  • ✓
    $f (x)$ is continuous at $x=0$
  • D
    $f (x)$ is not continuous at $x=0$
✓

Answer

Correct option: C.
$f (x)$ is continuous at $x=0$
(C)
$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)$
$=\lim _{h \rightarrow 0} \frac{-h}{e^{\frac{-1}{h}}+1}=\lim _{h \rightarrow 0} \frac{-h}{1+\frac{1}{e^{\frac{1}{h}}}}=0$
$\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0} f(0+h)=\lim _{h \rightarrow 0} \frac{h}{e^{\frac{1}{h}}+1}=0$
$\therefore \quad \lim _{x \rightarrow 0^{-}} f (x)=\lim _{x \rightarrow 0^{+}} f (x)= f (0)$
$\therefore \quad f (x)$ is continuous at $x=0$.

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