Let be defined by f(x)= cases 3x^2+4x-7 x-2 , & x<2

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

Let

f

be defined by

f(x)=\begin{cases} \dfrac{3x^2+4x-7}{x-2}, & x<2,\\ kx+1, & x\ge 2. \end{cases}

The constant

k

is unknown. (a) Determine the value of

k

that guarantees

\lim_{x\to 2} f(x)

exists. (b) For the value of

k

you found in part (a), evaluate

\lim_{x\to 2} f(x)

. Justify your work algebraically. (c) With the same

k

, determine whether

f

is continuous at

x=2

. Support your conclusion with appropriate reasoning. (d) For the

k

found above, evaluate

\lim_{x\to 2} (x-2)f(x)

and interpret the result in terms of the behavior of

f

near

x=2

. Explain your reasoning.