Mathematics

Problem

Let

g(x)=\begin{cases} 3x+k, & x<2,\\[4pt] x^2-3, & x\ge 2. \end{cases}

Determine the constant

k

that ensures

g

is continuous at

x=2

. Justify your answer with appropriate limit calculations. (a) Calculate the left–hand limit of

g

x

approaches

2

. (b) Calculate the right–hand limit of

g

x

approaches

2

. (c) Determine the value of

k

that makes the two one–sided limits equal, and state the common limit value.