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Continuity and Differentiability — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSContinuity and Differentiability1 Mark
Question
For which value of $k$, this funciton is continuous at $x=1$.$
f(x)=\left\{\begin{array}{cc}
\frac{x^2-3 x+2}{x-1}, & x \neq 1 \\
k, & x=1
\end{array}\right.
$
✓

Answer

$(B)$
value of function at $x=1$
$f(1)=k$
Value of $\text{R.H.L.}$
$\lim _{h \rightarrow 0} f(1+h)=\lim _{h \rightarrow 0} \frac{(1+h)^2-3(1+h)+2}{1+h-1}$
$=\lim _{h \rightarrow 0} \frac{1+2 h+h^2-3-3 h+2}{h}$
$=\lim _{h \rightarrow 0} \frac{h^2-h}{h}=\lim _{h \rightarrow 0}[h-1]=-1$
because function is continuous at $x=1$, so
$f(1)=\text{ R . H . L .}$
$\therefore k=-1$
Hence correct option is $(B).$

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