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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability1 Mark
MCQ
If function $f(x)=\left\{\begin{array}{c}\frac{x^2-16}{x-4}, x \neq 4 \\ k, x=4\end{array}\right.$, is continuous at $x=4$, then value of $k$ is :
  • A
    4
  • B
    -4
  • ✓
    8
  • D
    -8
✓

Answer

Correct option: C.
8
(C)value of function of $x=4$
$
f(4)=k
$
value of R.H.L. at $x=4$$
\begin{aligned}
\lim _{x \rightarrow 4^{+}} f(x) & =\lim _{h \rightarrow 0} f(4+h)=\lim _{h \rightarrow 0} \frac{(4+h)^2-16}{4+h-4} \\
& =\lim _{h \rightarrow 0} \frac{16+8 h+h^2-16}{h} \\
& =\lim _{h \rightarrow 0} \frac{h(8+h)}{h}=8
\end{aligned}
$
$\because$ function is continuous at $x=4$.
$\therefore \quad f(4)=\lim _{h \rightarrow 0} f(4+h)$
$\Rightarrow \quad k=8$

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