A) 3
B) 2
C) 1
D) -1
Correct Answer: A
Solution :
[a] Let \[f(x)=\left\{ \begin{matrix} \frac{{{x}^{3}}-3x+2}{{{(x-1)}^{2}}}, & \forall x\ne 1 \\ k, & \forall x=1 \\ \end{matrix} \right.\] and \[f(x)\] is continuous. \[\therefore \underset{x\to 1}{\mathop{\lim }}\,f(x)=k\] \[\Rightarrow \underset{x\to 1}{\mathop{\lim }}\,\frac{{{x}^{3}}-3x+2}{{{(x-1)}^{2}}}=k\] \[\Rightarrow k=\underset{x\to 1}{\mathop{\lim }}\,\frac{3{{x}^{2}}-3}{2(x-1)}[By\,\,L'\,\,Hospitals\,\,rule]\] \[\Rightarrow k=\underset{x\to 1}{\mathop{\lim }}\,\frac{6x}{2}[By\,\,L'\,\,Hospitals\,\,rule]\] \[\Rightarrow k=3\]
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