You will find a graphing calculator useful for Exercise. Let f(x)=(x^2-1) /(|x|-1). a. Make tabl

step 1 For the given problem, we want to find a graphing calculator. We're going to have f of x equalin

You will find a graphing calculator useful for Exercise. Let...

You will find a graphing calculator useful for Exercise. Let $f(x)=\left(x^{2}-1\right) /(|x|-1).$ a. Make tables of the values of $f$ at values of $x$ that approach $c=-1$ from above and below. Then estimate $\lim _{x \rightarrow-1} f(x).$ b. Support your conclusion in part (a) by graphing $f$ near $c=-1$ and using Zoom and Trace to estimate $y$ -values on the graph as $x \rightarrow-1.$ c. Find $\lim _{x \rightarrow-1} f(x)$ algebraically.

You will find a graphing calculator useful for Exercise.
Let $f(x)=\left(x^{2}-1\right) /(|x|-1).$
a. Make tables of the values of $f$ at values of $x$ that approach $c=-1$ from above and below. Then estimate $\lim _{x \rightarrow-1} f(x).$
b. Support your conclusion in part (a) by graphing $f$ near $c=-1$ and using Zoom and Trace to estimate $y$ -values on the graph as $x \rightarrow-1.$
c. Find $\lim _{x \rightarrow-1} f(x)$ algebraically.

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Key Concepts

Algebraic Simplification

Algebraic techniques such as factoring and canceling common factors are used to simplify expressions, eliminate indeterminate forms, and reveal the underlying behavior of a function near a certain point. This is critical in solving limits where direct substitution is not viable due to potential undefined terms.

Limit of a Function

This concept refers to the value that a function approaches as its input nears a particular point. It is a fundamental idea in calculus that helps to understand behavior near points where the function may not be explicitly defined, serving as a basis for discussing continuity and differentiability.

One-Sided Limits

One-sided limits involve evaluating the behavior of a function as the variable approaches a given point exclusively from either the left or the right side. This is especially important when the function has different expressions or behavior on different intervals or is defined piecewise.

Graphical Analysis

Graphing a function provides a visual understanding of its behavior near the point of interest. Tools like graphing calculators allow us to zoom in, trace the curve, and confirm the analytical results obtained by other methods, thus supporting conclusions drawn from algebraic manipulation and numerical estimation.

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