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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
Let $f(x) = \,\left\{ {\begin{array}{*{20}{c}}   {\frac{{\sin \pi x}}{{5x}},}&{x \ne 0} \\    {k,}&{x = 0}  \end{array}} \right.$ if $f(x)$  is continuous at  $x = 0,$  then $k=$
  • ✓
    $\frac{\pi }{5}$
  • B
    $\frac{5}{\pi }$
  • C
    $1$
  • D
    $0$
✓

Answer

Correct option: A.
$\frac{\pi }{5}$
a
(a) Since $f(x)$ is continuous at $x = 0,$ 

therefore $\mathop {{\rm{lim}}}\limits_{x \to 0} f(x) = f(0)$

==> $\mathop {{\rm{lim}}}\limits_{x \to 0} \frac{{\sin \pi \,x}}{{5x}} = k$

==> $\mathop {{\rm{lim}}}\limits_{x \to 0} \left( {\frac{{\sin \pi \,x}}{{\pi x}}} \right)\,.\,\frac{\pi }{5} = k$

==> $(1)\,.\,\frac{\pi }{5} = k$ ==> $k = \frac{\pi }{5}$.

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