Questions: Let f(x) = (x^2 - 1) / (x - 1) (a) Make tables of the values of f at values of x that approach x0 = -1 from above and below. Then estimate the limit as x approaches -1 of f(x). (b) Support your conclusion in part (a) by graphing f near x0 = -1 and using Zoom and Trace to estimate y-values on the graph as x approaches -1. (c) Find the limit as x approaches -1 of (x^2 - 1) / (x - 1) algebraically. (a) Complete the table given below.

Solution Steps

The function given is: \[ f(x) = \frac{x^2 - 1}{|x| - 1} \]

For \( x > 0 \): \[ f(x) = \frac{x^2 - 1}{x - 1} = \frac{(x - 1)(x + 1)}{x - 1} = x + 1 \]

For \( x < 0 \): \[ f(x) = \frac{x^2 - 1}{-x - 1} = \frac{(x - 1)(x + 1)}{-(x + 1)} = -(x - 1) = -x + 1 \]

For \( x \) approaching \(-1\) from above (\( x \to -1^+ \)): \[ f(x) = -x + 1 \] \[ f(-1^+) = -(-1) + 1 = 2 \]

For \( x \) approaching \(-1\) from below (\( x \to -1^- \)): \[ f(x) = -x + 1 \] \[ f(-1^-) = -(-1) + 1 = 2 \]

Since both approaches give the same value: \[ \lim_{{x \to -1}} f(x) = 2 \]

Final Answer