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Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity2 Marks
MCQ
Which of the following functions is continuous at $x=0$ ?
  • A
    $f(x)=\left\{\begin{array}{cl}\frac{\sin 2 x}{x} ; & x \neq 0 \\ 1 ; & x=0\end{array}\right.$
  • B
    $f(x)=\left\{\begin{array}{cl}(1+x)^{\frac{1}{x}} ; & x \neq 0 \\ 1 ; & x=0\end{array}\right.$
  • C
    $f (x)=\left\{\begin{array}{cc} e ^{\frac{-1}{x}} ; & x \neq 0 \\ 1 ; & x=0\end{array}\right.$
  • ✓
    $f(x)=\left\{\begin{array}{cc}\frac{3 x+4 \tan x}{x} ; & x \neq 0 \\ 7 ; & x=0\end{array}\right.$
✓

Answer

Correct option: D.
$f(x)=\left\{\begin{array}{cc}\frac{3 x+4 \tan x}{x} ; & x \neq 0 \\ 7 ; & x=0\end{array}\right.$
(D)
$\lim _{x \rightarrow 0} \frac{\sin 2 x}{x}=2 \neq f (0)$
$\lim _{x \rightarrow 0}(1+x)^{\frac{1}{x}}=e \neq f(0)$
$\lim _{x \rightarrow 0} e ^{\frac{-1}{x}}=\lim _{x \rightarrow 0} \frac{1}{ e ^{\frac{1}{x}}}=\frac{1}{ e ^{\infty}}=\frac{1}{\infty}=0 \neq f (0)$
$\lim _{x \rightarrow 0}\left(\frac{3 x}{x}+\frac{4 \tan x}{x}\right)=3+4=7= f (0)$
$\therefore f (x)$ is continuous at $x=0$.

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