Problem 12 - Precalculus: Mathematics for Calculus

Question

Estimating Limits Numerically and Graphically Use a table of values to estimate the value of the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 1} \frac{x^{3}-1}{x^{2}-1}$$

Step-by-Step Solution

Answer

The limit is approximately 2.

Step 1: Evaluate Function Values Near the Limit

We begin by evaluating the function \( f(x) = \frac{x^3 - 1}{x^2 - 1} \) at several points near \( x = 1 \). Create a table with values such as \( x = 0.9 \), \( x = 0.99 \), \( x = 1.01 \), \( x = 1.1 \): \[\begin{array}{c|c}\ x & f(x) \\hline0.9 & \frac{(0.9)^3 - 1}{(0.9)^2 - 1} \0.99 & \frac{(0.99)^3 - 1}{(0.99)^2 - 1} \1.01 & \frac{(1.01)^3 - 1}{(1.01)^2 - 1} \1.1 & \frac{(1.1)^3 - 1}{(1.1)^2 - 1} \\end{array}\]Compute these values to observe numerical trends.

Step 2: Calculate Values in the Table

Perform the calculations for each \( f(x) \) value listed in the table.\[f(0.9) = \frac{0.729 - 1}{0.81 - 1} \approx 1.9,\]\[f(0.99) = \frac{0.970299 - 1}{0.9801 - 1} \approx 2.009999,\]\[f(1.01) = \frac{1.030301 - 1}{1.0201 - 1} \approx 2.010101,\]\[f(1.1) = \frac{1.331 - 1}{1.21 - 1} \approx 2.1.\]These values suggest the limit as \( x \) approaches 1 is approximately 2.

Step 3: Analyze Trends in Calculated Values

Observing the computed values, as \( x \) gets closer to 1 from both sides, \( f(x) \) approaches a value near 2. Therefore, based on numerical estimation, we hypothesize that:\[\lim_{x \to 1} \frac{x^3 - 1}{x^2 - 1} \approx 2.\]

Step 4: Confirm with Graphing

Using a graphing device such as Desmos or a graphing calculator, plot the graph of \( y = \frac{x^3 - 1}{x^2 - 1} \). Observe the behavior of the graph as \( x \) approaches 1. You'll notice that as \( x \) approaches 1 from either direction, the value of \( y \) approaches 2, confirming our numerical estimate.

Key Concepts

Numerical Estimation of Limits

Numerical estimation is a technique used to approximate the value of a limit using a table of calculated function values near the point of interest. This approach is helpful when the function's algebraic manipulation might be challenging.
To estimate the limit of \( f(x) = \frac{x^3 - 1}{x^2 - 1} \) as \( x \) approaches 1, we can evaluate the function at points close to 1, such as 0.9, 0.99, 1.01, and 1.1.

Here's the general process:

• Pick numbers increasingly close to the point from both sides.
• Calculate the function value at each point.
• Look for a trend in the values.

By doing this, you find that as \( x \) gets nearer to 1, the function values approach 2, suggesting the limit is \( 2 \). This technique helps to gain confidence that our calculations are pointing towards the correct limit value.

Graphical Analysis of Limits

Graphical analysis provides a visual confirmation of the limit by observing the behavior of the function's graph near the point of interest.
Using graphing tools like Desmos or a graphing calculator, you can plot the function \( y = \frac{x^3 - 1}{x^2 - 1} \).

When analyzing the graph:

• Look closely at the function's behavior as \( x \) approaches the point from both directions.
• Note where the graph seems to settle as it nears the point of interest.
• Pay attention to its tendencies, particularly whether it heads towards a certain value.

For this function, as \( x \) approaches 1 from either side, the graph approaches a y-value of around 2. This visually confirms our numerical estimation of the limit value.

Function Behavior Near a Point

Understanding how a function behaves as it approaches a particular point is key to grasping the concept of limits. This analysis involves observing both the function's values and shape around the point.
The function \( f(x) = \frac{x^3 - 1}{x^2 - 1} \) is undefined at \( x = 1 \) due to a zero in the denominator, causing a "hole" in the graph.

Nevertheless, we can still explore:

• How the function's values change as \( x \) nears 1 from either direction.
• The mathematical simplification potential, identifying if a common factor can be canceled out.
• The tendency of the function values as they approach the point.

In this case, simplifying the expression reveals that both numerator and denominator share a factor of \( x-1 \): \[ \frac{x^3 - 1}{x^2 - 1} = \frac{(x-1)(x^2+x+1)}{x-1} = x^2 + x + 1 \] Thus, for \( x eq 1 \), the function behaves like \( x^2 + x + 1 \), which is smooth and continuous near \( x = 1 \), leading to the conclusion or understanding of behavior near this point.