Problem 17 - Precalculus: Mathematical for Calculus

Question

Use a graphing device to determine whether the limit exists. If the limit exists, estimate its value to two decimal places. $$\lim _{x \rightarrow 1} \frac{x^{3}+x^{2}+3 x-5}{2 x^{2}-5 x+3}$$

Step-by-Step Solution

Answer

The limit is \(-8\).

Step 1: Analyze the Function

The function given is \( f(x) = \frac{x^3 + x^2 + 3x - 5}{2x^2 - 5x + 3} \). We aim to find \( \lim_{x \to 1} f(x) \). First, analyze the denominator \( 2x^2 - 5x + 3 \) to ensure that it does not equal zero at \( x = 1 \).

Step 2: Evaluate the Denominator at x = 1

Calculate \( 2 \times 1^2 - 5 \times 1 + 3 \). This simplifies to \( 2 - 5 + 3 = 0 \). Since the denominator is 0 at \( x = 1 \), we need to analyze further to understand the behavior of the function near this point.

Step 3: Check the Numerator at x = 1

Evaluate the numerator \( x^3 + x^2 + 3x - 5 \) at \( x = 1 \). This gives \( 1 + 1 + 3 - 5 = 0 \). Since both the numerator and denominator are zero at \( x = 1 \), we have an indeterminate form \( \frac{0}{0} \).

Step 4: Simplification by Factoring

Factor both the numerator and the denominator if possible. The numerator \( x^3 + x^2 + 3x - 5 \) can be rewritten by factoring as \( (x - 1)(x^2 + 2x + 5) \) and the denominator \( 2x^2 - 5x + 3 \) can be factored as \( (x - 1)(2x - 3) \).

Step 5: Cancel the Common Factor

Cancel the common factor \( (x - 1) \) from both the numerator and the denominator. The function simplifies to \( \frac{x^2 + 2x + 5}{2x - 3} \).

Step 6: Evaluate the Simplified Limit

Substitute \( x = 1 \) into the simplified function \( \frac{x^2 + 2x + 5}{2x - 3} \): \( \frac{1^2 + 2 \times 1 + 5}{2 \times 1 - 3} = \frac{1 + 2 + 5}{2 - 3} = \frac{8}{-1} = -8 \).

Step 7: Estimate the Limit Using a Graphing Device

Use a graphing device to plot the function \( f(x) = \frac{x^3 + x^2 + 3x - 5}{2x^2 - 5x + 3} \). As \( x \) approaches 1, observe that the limit appears to be \( -8 \). This confirms the earlier algebraic solution.

Key Concepts

Indeterminate Forms

When dealing with limits, sometimes we encounter expressions that result in forms like \( \frac{0}{0} \), which are known as indeterminate forms. These forms do not immediately provide enough information to determine a limit. This is because dividing zero by zero doesn't produce a number or doesn’t fit the definition of standard arithmetic. Instead, it indicates that more work needs to be done such as simplification or another method of evaluation.

Indeterminate forms require us to look closer at the function involved. In the exercise example, evaluating both the numerator \( x^3 + x^2 + 3x - 5 \) and the denominator \( 2x^2 - 5x + 3 \) at \( x = 1 \) results in zero. Hence, it gave us the \( \frac{0}{0} \) form.

This implies we must continue analyzing possibly through algebraic manipulation like factoring or using other limit properties to gain insight or resolve the limit.

Factoring Algebraic Expressions

Factoring is an essential algebraic technique used to simplify expressions and solve equations. In the context of solving limits, factoring becomes crucial when dealing with indeterminate forms. By simplifying expressions, it often allows terms causing the indeterminate form to be canceled out or reduced.

For the given example, after recognizing that both the numerator and denominator equate to zero at \( x = 1 \), we proceed with factoring. The numerator \( x^3 + x^2 + 3x - 5 \) and denominator \( 2x^2 - 5x + 3 \) are factorized as \( (x - 1)(x^2 + 2x + 5) \) and \( (x - 1)(2x - 3) \) respectively.

Notice the common factor \( (x - 1) \) in both expressions. By canceling this common factor, the expression simplifies, allowing you to evaluate the limit directly by substitution without returning to an indeterminate form. Factoring is a powerful technique not only to simplify expressions but also to uncover meaningful relationships within equations.

Using Graphing Calculators

A graphing calculator is a valuable tool in calculus for visualizing functions and verifying analytical solutions, especially for limits. After finding the simplified expression through algebraic manipulation, a graphing calculator helps confirm the calculated limit by plotting the function.

Although algebraically we've computed that the limit of \( f(x) = \frac{x^2 + 2x + 5}{2x - 3} \) as \( x \) approaches 1 is -8, graphing allows us to see this behavior visually. By plotting, you observe how the values of \( f(x) \) behave as \( x \) nears 1.

This graphical representation is particularly beneficial when your algebraic solution involves complicated expressions or when you need to estimate solutions quickly. It also serves as a practical check to ensure the accuracy of manual calculations, providing a deeper understanding of function behavior near the point of interest.