You will find a graphing calculator useful for Exercise. Let f(x)=(x^2-9) /(x+3). a. Make a tabl

step 1 For the given problem, we're going to be using a graphene calculator to solve this particular pr

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

You will find a graphing calculator useful for Exercise. Let $f(x)=\left(x^{2}-9\right) /(x+3).$ a. Make a table of the values of $f$ at the points $x=-3.1$ $-3.01,-3.001,$ and so on as far as your calculator can go. Then estimate $\lim _{x \rightarrow-3} f(x) .$ What estimate do you arrive at if you evaluate $f$ at $x=-2.9,-2.99,-2.999, \ldots$ instead? b. Support your conclusions in part (a) by graphing $f$ near $c=-3$ and using Zoom and Trace to estimate $y$ -values on the graph as $x \rightarrow-3.$ c. Find $\lim _{x \rightarrow-3} f(x)$ algebraically, as in Example 7.

You will find a graphing calculator useful for Exercise.
Let $f(x)=\left(x^{2}-9\right) /(x+3).$
a. Make a table of the values of $f$ at the points $x=-3.1$ $-3.01,-3.001,$ and so on as far as your calculator can go. Then estimate $\lim _{x \rightarrow-3} f(x) .$ What estimate do you arrive at if you evaluate $f$ at $x=-2.9,-2.99,-2.999, \ldots$ instead?
b. Support your conclusions in part (a) by graphing $f$ near $c=-3$ and using Zoom and Trace to estimate $y$ -values on the graph as $x \rightarrow-3.$
c. Find $\lim _{x \rightarrow-3} f(x)$ algebraically, as in Example 7.

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Key Concepts

Graphical Analysis with Calculators

Using a graphing calculator to conduct graphical analysis allows one to visually inspect the behavior of a function near a specific point. By zooming in and tracing the function's curve, one can estimate limit values and confirm conclusions drawn from numerical tables and algebraic manipulation.

Algebraic Simplification

This involves factoring and canceling common terms in a rational expression to simplify it. Such simplification is key when evaluating limits, particularly in cases where direct substitution leads to indeterminate forms like 0/0. It reveals an equivalent form of the function that is easier to analyze near the point of interest.

Removable Discontinuity

A removable discontinuity occurs when a function is not defined at a point, but the limit exists because the problematic factor causing the discontinuity can be algebraically canceled. Recognizing and removing these discontinuities helps in accurately determining the function's behavior near that point.

Limit of a Function

This concept describes what value a function approaches as the input approaches a particular point, regardless of the function's value at that point. It's fundamental in calculus for understanding the behavior of functions near a point, and it underpins more advanced topics like continuity and derivatives.

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