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

You will find a graphing calculator useful for Exercises $67-76$. Let $f(x)=\left(x^{2}-9\right) /(x+3)$. 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 Exercises $67-76$.
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

Graphical analysis uses visual methods such as plotting the function, creating tables of values, and utilizing tools like zoom and trace on a graphing calculator to observe the behavior of a function near a point. This approach helps confirm the conclusions reached through algebraic techniques by providing an intuitive understanding of the limit.

Algebraic Simplification

Algebraic simplification involves techniques such as factoring and canceling common factors in the numerator and denominator of a function. This process is used to eliminate indeterminate forms like 0/0, thereby making it possible to directly evaluate the limit.

Limit

The concept of a limit involves determining the value that a function approaches as the input gets arbitrarily close to a particular point, even if the function is not defined at that point. It is a foundational idea in calculus used to analyze function behavior near points of interest.

Removable Discontinuity

A removable discontinuity occurs when a function is not defined at a certain point because of a factor that can be cancelled or simplified algebraically without changing the overall behavior of the function. Recognizing a removable discontinuity allows one to redefine the function or simplify the expression to evaluate the limit.

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