Estimating Limits


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


Let h(x)=(x² − 2x − 3)/(x² − 4x + 3)


b. Support your conclusions in part (a) by graphing h 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 h(x) = (x² − 2x − 3)/(x² − 4x + 3). This is a rational function, which means it is the ratio of two polynomials.
2. Identify any points of discontinuity by setting the denominator equal to zero: x² − 4x + 3 = 0. Solve this quadratic equation to find the values of x that make the denominator zero.
3. Factor the denominator: x² − 4x + 3 = (x - 3)(x - 1). The function h(x) is undefined at x = 3 and x = 1, which are the points of discontinuity.
4. To estimate the limit as x approaches 3, use a graphing calculator to plot the function h(x). Use the Zoom feature to focus on the area around x = 3.
5. Utilize the Trace feature on the graphing calculator to observe the behavior of the y-values as x approaches 3 from both sides. This will help you estimate the limit of h(x) as x approaches 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, understanding limits helps in analyzing the behavior of the function h(x) as x approaches 3, which is crucial for determining continuity and potential discontinuities in the function.

Graphing Functions

Graphing functions involves plotting the values of a function on a coordinate plane, which visually represents its behavior. In this exercise, using a graphing calculator to visualize h(x) near x = 3 allows for a better understanding of how the function behaves around that point, aiding in the estimation of limits.

Zoom and Trace Features

The Zoom and Trace features on a graphing calculator allow users to adjust the viewing window and track the values of a function as the input changes. This is particularly useful for estimating y-values as x approaches a specific point, such as 3 in this case, providing a practical method to support conclusions drawn from limit analysis.