Mathematics

Problem

Let the function

h

be defined by

h(x)=\begin{cases} \dfrac{2x^{2}-8}{x-2}, & x<2,\\ \ln(x-2), & x>2. \end{cases}

Investigate the limit behavior of

h

x=2

and determine the nature of any discontinuity. a. Part 1: (a) Algebraically simplify the expression

\dfrac{2x^{2}-8}{x-2}

for

x\ne2

, and state the simplified form. b. Part 2: (b) Compute

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

and

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

. Clearly indicate any use of the simplified form from part (a). c. Part 3: (c) Identify every vertical asymptote of

h

and give a brief justification for each one. d. Part 4: (d) Is

h

continuous at

x=2

? Classify the type of discontinuity that occurs, citing specific numerical evidence from part (b) and relevant limit definitions.