A function is defined by g(x)= cases x+1, & x<4 4x, & x 4 cases Which statement best…

MATHEMATICS · Single correct · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

A function

g

is defined by

g(x)=\begin{cases} x+1, & x<4\\ 4x, & x\ge 4 \end{cases}

Which statement best explains why

g

is not continuous at

x=4

? (a)

\displaystyle \lim_{x\to4} g(x)=5\neq g(4)

, so the unequal values make

g

discontinuous at

x=4

. (b) The limit of

g(x)

x

approaches

4

does not exist because the left-hand and right-hand limits are unequal. (c)

g(4)

is undefined, so the function cannot be continuous at

x=4

. (d) The right-hand limit equals

g(4)

while the left-hand limit differs, so the overall limit exists but continuity still fails.