Define the function by f(x)= cases (5-x), & x5

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Define the function ff by f(x)={ln⁡(5−x),x<5,x2−25x−5,x>5.f(x)=\begin{cases} \ln(5-x), & x<5,\\ \dfrac{x^{2}-25}{x-5}, & x>5. \end{cases} A student claims that ff possesses both a vertical asymptote and a removable discontinuity at x=5x=5. Investigate this claim thoroughly. (1) Part 1: (a) Compute lim⁡x→5−f(x)\displaystyle\lim_{x\to 5^-}f(x). Show analytical steps that justify your result. (2) Part 2: (b) Simplify x2−25x−5\dfrac{x^{2}-25}{x-5} for x≠5x\ne5, then compute lim⁡x→5+f(x)\displaystyle\lim_{x\to 5^+}f(x). (3) Part 3: (c) Based on parts (a) and (b), does lim⁡x→5f(x)\displaystyle\lim_{x\to 5}f(x) exist? Provide a formal justification. (4) Part 4: (d) Precisely classify the discontinuity of ff at x=5x=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. (5) Part 5: (e) Sketch (by verbal description) the graph of ff on the interval (2,8)(2,8). Your description should include: key intercepts, asymptotic behavior, continuity regions, and end behavior within the stated interval.

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