Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let f(x) = (x² - 9) / (x + 3)

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 → −3.

Guida verificata passo dopo passo

First, understand the function f(x) = (x² - 9) / (x + 3). Notice that the expression x² - 9 can be factored as (x - 3)(x + 3). This will help simplify the function.

Next, simplify the function by canceling out the common factor (x + 3) in the numerator and denominator, which gives f(x) = x - 3 for x ≠ -3. This simplification is valid except at x = -3, where the original function is undefined.

To estimate the limit as x approaches -3, use a graphing calculator to plot the function f(x) = (x² - 9) / (x + 3). Set the viewing window to focus around x = -3, such as from x = -4 to x = -2.

Utilize the Zoom feature on the graphing calculator to closely examine the behavior of the graph near x = -3. This will help you observe how the y-values change as x approaches -3 from both sides.

Finally, use the Trace feature to estimate the y-values of the graph as x approaches -3. This will provide a numerical approximation of the limit, supporting the conclusion that the limit of f(x) as x approaches -3 is the value of the simplified function at x = -3.

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Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, we are interested in the limit of f(x) as x approaches -3, which helps us understand the behavior of the function near that point, especially since direct substitution leads to an indeterminate form.

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Graphing Functions

Graphing functions allows us to visualize their behavior and identify key features such as intercepts, asymptotes, and continuity. By using a graphing calculator, we can observe how f(x) behaves as x approaches -3, which aids in estimating the limit and understanding the function's overall shape.

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Percorso guidato

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Graph of Sine and Cosine Function

Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit. In this case, substituting x = -3 into f(x) results in 0/0, prompting the need for further analysis, such as factoring or using graphical methods to resolve the limit.

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