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Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity2 Marks
MCQ
If $f(x)=\left\{\begin{array}{ll}\frac{5^x- e ^{ z }}{\sin 2 x} \quad ; \quad x \neq 0 \\ \frac{1}{2}(\log 5+1) \quad ; \quad x=0\end{array}\right.$ then
  • A
    $f (x)$ is continuous at $x=0$
  • ✓
    $f (x)$ is discontinuous at $x=0$
  • C
    $\lim _{x \rightarrow 0} f (x)$ does not exist
  • D
    none of these
✓

Answer

Correct option: B.
$f (x)$ is discontinuous at $x=0$
(B)
$\lim _{x \rightarrow 0} f (x)=\lim _{x \rightarrow 0} \frac{5^x- e ^x}{\sin 2 x}=\lim _{x \rightarrow 0} \frac{5^x-1+1- e ^x}{\sin 2 x}$
$=\lim _{x \rightarrow 0} \frac{\frac{5^x-1}{x}-\frac{ e ^x-1}{x}}{\frac{\sin 2 x}{2 x} \times 2}$
$=\frac{\log 5-\log e }{2}=\frac{1}{2}(\log 5-1)$
$\therefore \lim _{x \rightarrow 0} f (x) \neq f (0)$
$\therefore f (x)$ is discontinuous at $x=0$.

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