Problem 83 - Thomas Calculus

Question

Estimate the value of $$\lim _{x \rightarrow 1} \frac{2 x^{2}-(3 x+1) \sqrt{x}+2}{x-1}$$ by graphing. Then confirm your estimate with I'Hopital's Rule.

Step-by-Step Solution

Answer

The limit is 2.

Step 1: Set up the Problem for Graphing

We need to estimate the limit \( \lim_{x \rightarrow 1} \frac{2x^{2}-(3x+1)\sqrt{x}+2}{x-1} \) by graphing. To do this, create a graph of the function \( f(x) = \frac{2x^2 - (3x+1)\sqrt{x} + 2}{x-1} \) using a graphing calculator or software. Focus particularly on the behavior of the graph as \( x \) approaches 1 from both the left and the right.

Step 2: Analyze the Graph

After graphing the function, look at the behavior of the graph as it gets closer to \( x = 1 \). Typically, the y-value of the function as \( x \) approaches 1 from both sides should provide an estimate of the limit. Check if the function appears to stabilize at a particular y-value around \( x = 1 \).

Step 3: Confirm with I'Hôpital's Rule

To confirm the limit by using I'Hôpital's Rule, first verify that the given limit is an indeterminate form \( \frac{0}{0} \). Substitute \( x = 1 \) into both the numerator and denominator of \( f(x) \): both the numerator, \( 2(1)^2 - (3(1)+1)\sqrt{1} + 2 = 0 \), and the denominator, \( 1 - 1 = 0 \). Next, differentiate the numerator and the denominator separately: \( f'(x) = \frac{d}{dx}[2x^2-(3x+1)\sqrt{x}+2] \) and \( g'(x) = \frac{d}{dx}[x - 1] \). Apply I'Hôpital's Rule by finding the limit \( \lim_{x \rightarrow 1} \frac{f'(x)}{g'(x)} \).

Step 4: Differentiate the Numerator and Denominator

Calculate the derivative of the numerator: \( 4x - \frac{3}{2\sqrt{x}} - \frac{1}{2} \) and the derivative of the denominator: 1. Now apply I'Hôpital's Rule: \( \lim_{x \rightarrow 1} \frac{4x - \frac{3}{2\sqrt{x}} - \frac{1}{2}}{1} \).

Step 5: Evaluate the Limit of Derivatives

Substitute \( x = 1 \) into the differentiated expression: \( 4(1) - \frac{3}{2\sqrt{1}} - \frac{1}{2} = 4 - \frac{3}{2} - \frac{1}{2} = 2 \). Thus, the limit is \( 2 \), confirming the estimate made from the graph.

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule provides a method for evaluating limits of a form that appear as indeterminate ratios, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). This rule simplifies the process of finding limits by using derivatives.
When facing an indeterminate form, the process involves:

• Verifying that the initial limit results in an indeterminate form.
• Taking the derivative of the numerator and the derivative of the denominator separately.
• Re-evaluating the limit using the derivatives: \( \lim_{{x \to c}} \frac{f'(x)}{g'(x)} \).

In our exercise, when substituting \( x = 1 \) into \( \lim_{{x \rightarrow 1}} \frac{2x^2 - (3x+1)\sqrt{x} + 2}{x-1} \), the zero in the numerator and the denominator confirms the \( \frac{0}{0} \) indeterminate form.
The next step involves differentiating, starting with the numerator \( 4x - \frac{3}{2\sqrt{x}} - \frac{1}{2} \) and the denominator, which is simply 1. Finally, substituting \( x = 1 \) into these derivatives gives us a definitive limit of 2.
This process shows the power of L'Hôpital's Rule for simplifying seemingly complex limits.

Graphical Estimation of Limits

Graphical estimation of limits involves visually confirming the value that a function approaches as the variable closes in on a given point. This method is useful for gaining an intuitive understanding of the behavior of a function.
To graphically estimate a limit:

• Construct a plot of the function using graphing tools, paying close attention to the region surrounding the point of interest.
• Analyze the plot to observe how the function behaves as it nears the specified point from both the left and right sides.
• Seek a pattern in the y-values, as this can signify where the function is heading.

In our case, we graph the function \( f(x) = \frac{2x^2 - (3x+1)\sqrt{x} + 2}{x-1} \) and specifically watch its behavior as \( x \) approaches 1. The estimation becomes apparent if the graph stabilizes near a certain y-value. Here, this y-value approximation is 2, hinting at the limit.
This graphical method lays the groundwork, enabling a preliminary check before any analytical calculations.

Derivative Calculation

Calculating derivatives is an essential skill in calculus, particularly when applying rules like L'Hôpital's for limits. Derivatives represent the rate of change of a function with respect to a variable.
In the context of our problem, finding the derivative of both the numerator and denominator separately is a key step in using L'Hôpital's Rule. Let's examine why:

• The derivative of the numerator \( 2x^2 - (3x+1)\sqrt{x} + 2 \) requires using rules such as the power rule and chain rule. The result is \( 4x - \frac{3}{2\sqrt{x}} - \frac{1}{2} \).
• The denominator, a simple linear expression \( x-1 \), has a derivative of 1.

This differentiation reveals how the limit shifts from an indeterminate form to a calculable value after substitution. By knowing how to effectively find derivatives, the solution with L'Hôpital's Rule becomes straightforward and reliable. Mastery of this technique adds rigor and precision to your calculus arsenal.