Mathematics

Problem

Define the function

f

f(x) = \begin{cases} \dfrac{x^{2}-4}{x-2}, & x<2, \\ |x-2|+a, & 2\le x<5, \\ \ln(x-4), & x\ge 5, \end{cases}

where

a

is a real constant. (a) Evaluate

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

. (b) Evaluate

\displaystyle \lim_{x\to 2^{+}} f(x)

in terms of

a

. (c) Find the value of

a

that guarantees

f

is continuous at

x=2

and justify your result. (d) Using your value of

a

, state all intervals on which

f

is continuous and explain how you arrived at each interval.