Problem 14 - Precalculus with Limits

Question

In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically. $$\lim_{x \to -3} \dfrac{2x^2+5x-3}{x+3}$$

Step-by-Step Solution

Answer

The limit as x approaches -3 is -7.

Step 1: Factor the Polynomial in the Numerator

Express the numerator \(2x^2+5x-3\) as \((2x-1)(x+3)\). Thus, the expression becomes \[\lim_{x \to -3} \dfrac{(2x-1)(x+3)}{(x+3)}.\]

Step 2: Simplify the expression

The factor causing the undefined form at \(x =-3\) is \(x+3\). Cancel out this factor: \[\lim_{x \to -3} 2x -1.\] Now the expression is no longer undefined at \(x = -3\).

Step 3: Substitute the x-value

Substitute \(x = -3\) in the expression \((2x - 1)\) to get the limit. The resulting value is -7.

Step 4: Verify Result graphically

First, plot the function \(y=(2x-1)\). Then draw a vertical line at \(x=-3\). This will visualize the limit as \(x\) approaches -3. The y-value where the line intersects the function is the limit. If the y-value equals -7, the limit has been successfully verified.

Key Concepts

Polynomial Factoring

Polynomial factoring is the process of breaking down a polynomial into simpler components called factors. This method is essential when simplifying expressions, like rational functions, to find limits, especially when the expression initially appears undefined for certain values of \(x\). In this example, the expression \(2x^2 + 5x - 3\) needs to be factored to simplify the rational function.

• The polynomial \(2x^2 + 5x - 3\) is a quadratic, meaning it can potentially be factored into two binomials.
• Through factoring, the numerator \(2x^2+5x-3\) is expressed as the product \((2x-1)(x+3)\).

This step is crucial because it reveals common factors with the denominator, which can then be canceled out to simplify the expression. Factoring makes it possible to handle problematic points, such as those causing undefined values in rational functions.

Rational Functions

Rational functions are expressions that involve ratios of polynomials. Understanding these is key to finding limits, especially near points that cause problematic polynomial divisions.

• For the function \(\frac{2x^2+5x-3}{x+3}\), the goal is to simplify the function around the point \(x = -3\).
• The expression initially appears to be undefined at \(x = -3\) because the denominator becomes zero.

After factoring the numerator as \((2x-1)(x+3)\), we are able to cancel the \((x+3)\) terms:

• It simplifies the rational function to just \((2x-1)\) without the undefined factor.
• This transformation allows the limit to be evaluated straightforwardly by substituting \(x = -3\) into the simplified expression.

The ability to identify and cancel such factors helps avoid indeterminate forms and find the limit directly.

Graphical Verification

Graphical verification is a technique used to confirm analytic solutions using visual aids. Graphing helps illustrate the behavior of functions at specific points where limits are evaluated.

• In the solution, after simplifying the function to \(y = 2x - 1\), graphing can verify the result.
• By plotting the function and drawing a vertical line at \(x = -3\), you can observe where the line intersects the graph.

This intersection point gives a visual representation of the limit:

• The y-value at this intersection should match the calculated limit, \(-7\), confirming the calculation.

Graphical verification helps reinforce understanding by showing the correspondence between algebraic solutions and the actual behavior of the function, making abstract concepts like limits more tangible.