You will find a graphing calculator useful. Let f(x)=(x^2-9) /(x+3) a. Make a table of the value

step 1 Okay, so if you were to find the limits as x approaches negative 3 of f of x, let's see. So we'r

You will find a graphing calculator useful. Let f(x)=(x^2-9)...

You will find a graphing calculator useful. 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.
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 involves visualizing the function to observe its behavior near certain points. By using tools like zoom and trace functions on a graphing device, one can estimate limits by seeing how the function values change as the input approaches the point of interest, providing an intuitive understanding of the limit concept.

Algebraic Simplification

Algebraic simplification involves manipulating expressions to reveal underlying structures, such as factoring and canceling common terms. This technique is particularly useful in limit problems where direct substitution results in an indeterminate form, as it can simplify the expression to one that can be evaluated directly.

Removable Discontinuities

A removable discontinuity occurs when a function is not defined at a particular point but can be 'fixed' by redefining the function's value at that point. In many cases, the function simplifies to a form that is continuous at that point, allowing us to define the limit there even if the function itself is undefined at the specific value.

Limits

Limits describe the behavior of functions as their input approaches a particular value. They are a foundational concept in calculus, providing a way to analyze how functions behave near points of interest, even where the function may not be explicitly defined.

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