Mathematics

Problem

Consider the function

f(x)=\begin{cases} 3x-5, & x<2,\\ a(x-2)^2+7, & x\ge 2. \end{cases}

where

a

is a real constant. Determine the value of

a

such that

f

is continuous at

x=2

. Show all work and clearly justify your reasoning using the formal definition of continuity at a point. a. Part 1: (a) Evaluate

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

and

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

in terms of

a

. b. Part 2: (b) Using your results from part (a), determine the value of

a

for which

f

satisfies the condition required for continuity at

x=2

. Explain how each of the three clauses in the definition of continuity is met.