You will find a graphing calculator useful for Exercises...

You will find a graphing calculator useful for Exercises $67-76$. 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 Exercises $67-76$.
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.

Key Concepts

Algebraic Simplification

Algebraic manipulation, including factoring and cancelation, is a key strategy for evaluating limits. Simplifying expressions can help in eliminating indeterminate forms like 0/0, making it possible to directly compute the limit and understand the underlying behavior of the function.

Removable Discontinuity

A removable discontinuity is a point at which a function is not defined or does not match its limit, even though the limit exists from both sides. By simplifying the algebraic expression, such as canceling common factors, one can often reveal a continuous function at that point, indicating that the discontinuity could be 'removed' by appropriately defining the function at the problematic point.

One-Sided Limits

One-sided limits focus on the behavior of a function as the input approaches the target value from one side—either from above (right-hand limit) or from below (left-hand limit). Assessing one-sided limits is particularly useful when the function exhibits different behavior on either side of a point, such as near discontinuities or points where the function may not be continuous.

Limit of a Function

The concept of a limit involves determining the value that a function approaches as the input approaches a certain point. It is a fundamental idea in calculus that helps us understand the behavior of functions near points of interest, even if the function is not explicitly defined at these points.

Absolute Value and Piecewise Functions

Absolute value functions often require special consideration because they are defined piecewise, resulting in different expressions depending on whether the input is positive or negative. This piecewise nature may necessitate separate analysis over different intervals to correctly evaluate limits or other properties of the function.

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