Problem 71 - Thomas Calculus

Question

You will find a graphing calculator useful. Let $$f(x)=\left(x^{2}-1\right) /(|x|-1).$$ a. Make tables of the values of \(f\) at values of \(x\) that approach \(x_{0}=-1\) from above and below. Then estimate \(\quad \lim _{x \rightarrow-1} f(x).\) b. Support your conclusion in part (a) by graphing \(f\) near \(x_{0}=-1\) and using Zoom and Trace to estimate \(y\) -values on the graph as \(x \rightarrow-1\) . c. Find \(\lim _{x \rightarrow-1} f(x)\) algebraically.

Step-by-Step Solution

Answer

The limit \( \lim_{x \rightarrow -1} f(x) \) is \(-2\).

Step 1: Understand the Function

The function given is \( f(x)=\left(x^{2}-1\right) /(|x|-1) \). Notice that both the numerator \( x^2 - 1 \) and the denominator \( |x|-1 \) can be zero at \( x = -1 \). This might indicate an indeterminate form when evaluating the limit.

Step 2: Table of Values from the Right

Create a table of \( f(x) \) values for \( x \) slightly greater than \(-1\), e.g., \(-0.9, -0.99, -0.999\). Calculate the corresponding \( f(x) \) values and observe their trend as \( x \) approaches \(-1\) from the right.

Step 3: Table of Values from the Left

Create another table of \( f(x) \) values but for \( x \) values slightly less than \(-1\), e.g., \(-1.1, -1.01, -1.001\). Calculate the \( f(x) \) values and determine their trend as \( x \) approaches \(-1\) from the left.

Step 4: Estimating the Limit from Tables

Analyze the tables from Steps 2 and 3. If the \( f(x) \) values from both sides tend towards a specific value as \( x \) approaches \(-1\), this is an estimation for \( \lim_{x \rightarrow -1} f(x) \).

Step 5: Graphing the Function

Use a graphing calculator to plot \( f(x) \) near \( x_0 = -1 \). Utilize the Zoom feature to closely inspect the graph near \( x = -1 \) and the Trace function to help determine \( f(x) \) values as \( x \rightarrow -1 \).

Step 6: Analyzing the Graph

Identify any potential discontinuities or areas of interest on the graph as \( x \rightarrow -1 \). Confirm if the graph supports the limit estimated from the tables, signalling the trend towards a particular value.

Step 7: Algebraic Manipulation

Simplify the expression \( \frac{x^2 - 1}{|x|-1} \). Notice \( x^2 - 1 = (x+1)(x-1) \) and adjust for \( |x| \): - If \( x > 0 \): \( |x| = x \), so \( \frac{x^2-1}{x-1} \) simplifies to \( x+1 \).- If \( x < 0 \): \( |x| = -x \), so \( \frac{x^2-1}{-x-1} \) simplifies to \( -(x+1) \).Since we are concerned with \( x \rightarrow -1 \), utilize the representation \( f(x) = -2 \) for values close to \(-1\) from both sides.

Step 8: Finding the Limit

Given the consistent behavior observed for \( f(x) \) approaching \(-2\) from both tables and the graph, conclude \( \lim_{x \rightarrow -1} f(x) = -2 \).

Key Concepts

Indeterminate Forms

In calculus, an indeterminate form is an expression that does not directly provide enough information to determine its true value, often encountered when evaluating limits. These forms arise when both the numerator and the denominator of a function approach zero as the variable approaches a certain point. For the function \( f(x) = \frac{x^2 - 1}{|x| - 1} \), both the numerator \( x^2 - 1 \) and the denominator \( |x| - 1 \) become zero at \( x = -1 \). This results in the indeterminate form \( \frac{0}{0} \), which typically requires additional techniques to evaluate the limit correctly. To resolve indeterminate forms, we often simplify the expression by factoring or using L'Hôpital's Rule if applicable, though in this exercise, algebraic simplification helped clarify the limit.

Graphical Analysis

Graphical analysis involves using visual tools like graphs to understand the behavior of functions, particularly to estimate limits and identify discontinuities. When graphing \( f(x) = \frac{x^2 - 1}{|x| - 1} \) near \( x = -1 \), plotting involves noticing trends and behaviors, utilizing graphing calculators' Zoom and Trace features.

• The Zoom function allows detailed inspection near points of interest, such as \( x = -1 \), potentially revealing behaviors not evident at broader scales.
• The Trace function aids in tracking values of \( f(x) \) as \( x \) nears \(-1\), supporting estimates of the limit by observing consistent trends.

By analyzing the graph, one may confirm the value of the limit, as confrontations around \( x = -1 \) should consistently trend towards the estimated limit value. Such analysis complements algebraic techniques by offering a visual confirmation or illustration of a function's behavior.

Piecewise Functions

Piecewise functions are those defined by different expressions depending on the input value intervals. The modulus function involved in \( f(x) = \frac{x^2 - 1}{|x| - 1} \) results in a piecewise nature of \( f(x) \) when considering the absolute value definition. Determining limits near critical points often requires analyzing separate expressions for different intervals:

• For \( x > 0 \), the absolute value \( |x| = x \) applies, simplifying \( f(x) \) to \( \frac{x^2 - 1}{x - 1} \).
• For \( x < 0 \), the absolute value becomes \( |x| = -x \), leading to a different expression \( \frac{x^2 - 1}{-x - 1} \).

When \( x \) approaches \(-1\), the function simplifies and behaves consistently, regardless of approaching from left or right. This consistency in a piecewise function confirms the calculated limit, hence showcasing how different segments of a piecewise function can converge on the same limit, even when defined by distinct expressions.