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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability1 Mark
MCQ
If function $f(x)=\left\{\begin{array}{l}1+x, x \leq 2 \\ k-\frac{x}{2}, x>2\end{array}\right.$, is continuous at $x=2$, then value of $k$ will be :
  • A
    1
  • B
    2
  • C
    3
  • ✓
    4
✓

Answer

Correct option: D.
4
(D)value of function at $x=2$$
\begin{aligned}
f(2) & =1+2=3 \\
f(2+h) & =k-\left(\frac{2+h}{2}\right) \\
f(2-h) & =1+(2-h)=(3-h)
\end{aligned}
$
Value of R.H.L.$
\begin{aligned}
f(2+0)=\lim _{h \rightarrow 0} f(2+h) & =\lim _{h \rightarrow 0}\left[k-\left(\frac{2+h}{2}\right)\right] \\
& =k-1
\end{aligned}
$
Value of L.H.L.$
f(2-0)=\lim _{h \rightarrow 0} f(2-h)=\lim _{h \rightarrow 0}[3-h]=3
$
Function is continuous at $x=2$, so.$
\begin{aligned}
& & f(2) & =\text { R.H.L. }=\text { L.H.L. } \\
\Rightarrow & & 3 & =k-1=3 \therefore k=4
\end{aligned}
$
Hence correct option is (D).

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