Consider the function f(x)= cases x^ 2 -1 x-1 , & x 1, 3, & x=1

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

Problem

Consider the function f(x)={x2−1x−1,x≠1,3,x=1.f(x)=\begin{cases} \dfrac{x^{2}-1}{x-1}, & x\neq 1,\\ 3, & x=1. \end{cases} Use the information below to analyze the behavior of ff near x=1x=1. (1) Part 1: Calculate lim⁡x→1f(x)\displaystyle\lim_{x\to1} f(x), showing all algebraic steps. (2) Part 2: Explain in complete sentences why the value obtained in part (a) is consistent with simplifying the expression for f(x)f(x) when x≠1x\neq1. In your explanation, reference the formal definition of a limit. (3) Part 3: Determine whether ff is continuous at x=1x=1 and justify your conclusion.

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