Mathematics

Problem

Consider the function

f(x)=\begin{cases} mx^2+3, & x<2,\\ 7, & x=2,\\ \sqrt{x+5}, & x>2, \end{cases}

where

m

is a real constant. a. Part 1: (a) Determine

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

. b. Part 2: (b) Determine

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

. c. Part 3: (c) Find the value of

m

(if any) that guarantees

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

exists. d. Part 4: (d) For the value of

m

found in part (c), decide whether

f

is continuous at

x=2

. Justify your conclusion by explicitly referencing the definition of continuity.