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.

Verified step by step guidance

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.

Key Concepts

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.

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.

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.