Problem 45 - Precalculus with Limits

Question

In Exercises 37-48, use a graphing utility to graph the function and approximate the limit accurate to three decimal places. $$\lim_{x \to 1} \dfrac{1- \sqrt[3]{x}}{1-x}$$

Step-by-Step Solution

Answer

The limit is approximately -0.500, when x approaches 1.

Step 1 Graph the function

Begin by graphing the function \( y = \frac{1- \sqrt[3]{x}}{1-x} \). For this, a graphing utility is required. Plot the function in the graph and look at value as \( x \) approaches 1.

Step 2 Approximate the limit

Now, examine the graph closely and find the value of \( y \) as \( x \) gets very close to 1. This is the limit. Make sure the approximation is up to three decimal places, as requested in the question.

Step 3 Double check

To keep the accuracy of the answer, it's advised to double check if the limit value remains consistent for both as \( x \) approaches 1 from the left and right.

Key Concepts

Graphing Functions

Graphing functions is a visual approach to understanding how different variables relate to each other in mathematical equations. Working with a function like \( y = \frac{1- \sqrt[3]{x}}{1-x} \), the first step is to accurately plot the graph. A graphing utility, which can be a specialized calculator or software, helps us visualize this function. When graphing, look for key characteristics such as symmetry, intercepts, and particularly the behavior of the function near interesting points such as where \( x \to 1 \).

Here are a few helpful steps for graphing:

• Identify the domain of the function. For this function, check the values that make the denominator zero.
• Plot key points and observe the curve's path as \( x \) approaches the point of interest.
• Use a fine scale to observe changes around the point where the limit is sought.

Visualization can make abstract concepts more tangible by depicting how the function behaves as \( x \) varies, which is particularly useful for calculating limits.

Approximating Limits

Approximating limits is determining what value a function approaches as the input (\( x \)) closes in on a specific point. In the exercise, this means observing the graph of the function as \( x \to 1 \). Limits describe what happens at particular points, especially when the function isn't easily evaluated there due to indeterminate forms like \( \frac{0}{0} \).

This is how you can approximate limits:

• On the graph, closely observe the y-values as \( x \) nears 1.
• Take note of limits from both sides, left (\( x \to 1^-\)) and right (\( x \to 1^+\)), to verify if they approach the same value consistently.
• Record this limit to three decimal places as required for accuracy.

Approximating limits ensures a precise understanding of the function's behavior, especially around critical points like where discontinuities or undefined expressions might arise.

Graphing Utilities

Graphing utilities are powerful tools that enhance our ability to analyze and understand complex functions. They provide visual representations of equations and assist in identifying patterns, intercepts, and limits. Software packages such as Desmos or graphing calculators like the TI-84 allow for instant graph plotting and limit examination.

Here's why graphing utilities are beneficial:

• They quickly generate precise graphs of complicated functions.
• They allow zooming into regions of interest to inspect limit behavior at specific points.
• They offer additional features such as derivative visualization, integral approximation, and function transformation.

By using graphing utilities, the tedious arithmetic and manual plotting are eliminated, providing clarity and facilitating deeper insights into the function's nature which are crucial for tasks like estimating limits.