Question

$Letf\left(x\right)=\{\begin{array}{cc}{x}^2& {}_{\frac{k(x^2-4)}{2-x}}\end{array}\text{when x is an int eger otherwise then}{lim}_{x\to 2}f\left(x\right)$

• A. exists only when k=1
• B. exists for every real k
• C. exists for every real k =1
• D. does not exist

Answer 1

The correct option is B

exists for every real k

${lim}_{x\to 2^{+}}f\left(x\right)={lim}_{x\to 2^{+}}\frac{k(x^2-4)}{2-x}={lim}_{x\to 2^{+}}k(-2-x)=-4k{lim}_{x\to 2^{+}}f\left(x\right)={lim}_{x\to 2^{+}}\frac{k(x^2-4)}{2-x}={lim}_{x\to 2^{-}}k(-2-x)=-4k$

Hence limits exists for every real value of k.