Mathematics

Problem

Define the function

f

f(x)=\begin{cases} \ln(5-x), & x<5,\\ \dfrac{x^{2}-25}{x-5}, & x>5. \end{cases}

A student claims that

f

possesses both a vertical asymptote and a removable discontinuity at

x=5

. Investigate this claim thoroughly. a. Part 1: (a) Compute

\displaystyle\lim_{x\to 5^-}f(x)

. Show analytical steps that justify your result. b. Part 2: (b) Simplify

\dfrac{x^{2}-25}{x-5}

for

x\ne5

, then compute

\displaystyle\lim_{x\to 5^+}f(x)

. c. Part 3: (c) Based on parts (a) and (b), does

\displaystyle\lim_{x\to 5}f(x)

exist? Provide a formal justification. d. Part 4: (d) Precisely classify the discontinuity of

f

x=5

by stating whether it is removable, infinite (essential), or jump, and defend your classification with limit-based reasoning. Include any equations of vertical asymptotes that arise. e. Part 5: (e) Sketch (by verbal description) the graph of

f

on the interval

(2,8)

. Your description should include: key intercepts, asymptotic behavior, continuity regions, and end behavior within the stated interval.

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