Limits Continuity And Differentiability Question 62
Question: What is the value of k for which the following function f(x) is continuous for all x? $ f(x)={ \begin{aligned} & \frac{x^{3}-3x+2}{{{(x-1)}^{2}}},forx\ne 1 \\ & k,forx=1 \\ \end{aligned} }. $
Options:
A) 3
B) 2
C) 1
D) -1
Show Answer
Answer:
Correct Answer: A
Solution:
Let $ f(x)= \begin{matrix} \frac{x^{3}-3x+2}{{{(x-1)}^{2}}}, & \forall x\ne 1 \\ k, & \forall x=1 \\ \end{matrix} . $ 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{3x^{2}-3}{2(x-1)}[ByL’Hospitalsrule] $
$ \Rightarrow k=\underset{x\to 1}{\mathop{\lim }},\frac{6x}{2}[ByL’Hospitalsrule] $
$ \Rightarrow k=3 $
BETA
