Problem 49 - Precalculus with Limits

Question

GRAPHICAL, NUMERICAL, AND ALGEBRAIC ANALYSIS In Exercises 49-54, (a) graphically approximate the limit (if it exists) by using a graphing utility to graph the function, (b) numerically approximate the limit (if it exists) by using the \(table\) feature of a graphing utility to create a table, and (c) algebraically evaluate the limit (if it exists) by the appropriate technique(s). $$\lim_{x \to 1^-} \dfrac{x-1}{x^2 -1}$$

Step-by-Step Solution

Answer

The limit of the function \( \dfrac{x-1}{x^2 -1} \) as \( x \) approaches 1 from the left is 0.5.

Step 1 - Graphical Analysis

First, graph the function \( \dfrac{x-1}{x^2 -1} \) using a graphing utility. Pay close attention to the behavior of the function as \( x \) approaches 1 from the left (negative side). The limit is the y-value that the function is getting closer to as \( x \) approaches 1 from the left.

Step 2 - Numerical Analysis

Use the table feature of a graphing utility to create a table of values for the function as \( x \) approaches 1 from the left. Choose numerical values for \( x \) that are less than 1 and get very close to 1 (like 0.9, 0.99, 0.999, etc). Compute the corresponding y-values. As \( x \) gets closer and closer to 1, the y-values should be approaching the same value. This value will be the numerical approximation of the limit.

Step 3 - Algebraic Analysis

Finally, use algebraic techniques to evaluate the limit. First, observe that plugging \( x = 1 \) into the function yields \( \dfrac{1-1}{1^2 -1} = \dfrac{0}{0} \), this is an indeterminate form. So, use the algebraic technique of factoring. The function \( \dfrac{x-1}{x^2 -1} \) can be factored as \( \dfrac{x-1}{(x-1)(x+1)} \). Now, the limit can be found by canceling out the \( x-1 \) terms and allowing \( x \) to approach 1: \( \lim_{x \to 1^-} \dfrac{x-1}{(x-1)(x+1)} = \lim_{x \to 1^-} \dfrac{1}{x+1} = \dfrac{1}{1+1} = 0.5 \)

Key Concepts

Graphical Analysis

Graphical analysis of a function involves observing the behavior of the function on a graph. This can help us understand the limit a function approaches near specific points. To analyze the limit of the function \( \frac{x-1}{x^2 -1} \) as \( x \) approaches 1 from the left, you would graph this function. Make sure to zoom into the region near \( x = 1 \).

As \( x \) gets closer to 1 from values less than 1, observe the y-values on the graph. This is particularly useful to visualize how the function behaves as it approaches a certain point. If the y-value is approaching a specific number, this value is the graphical approximation of the limit.

Graphical analysis is valuable because it provides a visual representation of the function's behavior, helping to intuitively establish whether a limit exists and approximately what it might be.

Numerical Analysis

In numerical analysis, we use specific values to estimate the limit of a function. This approach involves checking the function's values at points that are very close to the point of interest. By using the table feature of a graphing utility, you can create a table of values for \( \frac{x-1}{x^2-1} \) as \( x \) nears 1 from the left.

Select values such as 0.9, 0.99, 0.999, and compute the function's corresponding y-values. As you notice these values, if they begin to converge or get extremely close to a particular number, then this number would be the numerical approximation of the limit.

• Choose \( x \) values less than 1 but very close to it.
• Observe the y-values as they stabilize around a particular value.

Numerical analysis provides solid evidence that supplements graphical observations, offering a practical and computational way to approximate limits.

Algebraic Techniques

Algebraic techniques offer a precise method for finding limits analytically. When evaluating \( \lim_{x \to 1^-} \frac{x-1}{x^2-1} \), substituting \( x = 1 \) results in an indeterminate form \( \frac{0}{0} \). To resolve this, we use factoring. By factoring \( x^2 - 1 \) into \( (x-1)(x+1) \), the expression becomes \( \frac{x-1}{(x-1)(x+1)} \).

With like terms in the numerator and denominator, you can cancel \( x-1 \), evaluating the limit as \( \lim_{x \to 1^-} \frac{1}{x+1} \). Now with this simpler form, directly substitute \( x = 1 \), yielding \( \frac{1}{2} \).

• Identify when a direct substitution leads to an indeterminate form.
• Factor and simplify expressions to remove problematic terms.
• Calculate the limit using the simplified form.

Algebraic techniques allow for finding the exact limit, showing why and how the limit equals \( 0.5 \) in this case.