Consider the function f(x)= cases 2x+3,& x<1

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Consider the function

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

The constant

k

is a real number. The behavior of

f

near

x=1

is of particular interest. (a) Determine

\displaystyle \lim_{x\to 1^-}f(x)

and

\displaystyle \lim_{x\to 1^+}f(x)

in terms of

k

. Show all algebraic steps. (b) When

k=0

, classify the type of discontinuity that occurs at

x=1

. Fully justify your classification by referencing the limits found in part (a) and the function value

f(1)

. (c) For

k=0

, compute the exact jump size at

x=1

. Provide clear reasoning. (d) Find the value of

k

that makes

f

continuous at

x=1

. Explain how your result follows from the formal definition of continuity.