Problem 8 - Precalculus with Limits

Question

In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached. $$\lim_{x \to 1}\ (2x^2+x-4)$$

Step-by-Step Solution

Answer

The limit of the function as \(x\) approaches \(1\) is -1.

Step 1: Substitute \(x\) with 1

Substitute \(x = 1\) into the function \(2x^2+x-4\) to get \(2(1)^2+1-4\).

Step 2: Simplify the expression

Simplify the expression by first calculating the power of \(1\), which remains \(1\), and then performing the addition and subtraction in the correct order. Multiplications before addition or subtraction: First calculate \(2*1 = 2\), resulting in \(2+1-4\). Now perform the addition and subtraction, resulting in -1.

Step 3: Review the result

Since substituting \(x = 1\) into the function resulted in a real number (-1), this number is the limit of the function as \(x\) approaches \(1\).

Key Concepts

Numerical estimation

Numerical estimation is a technique often used in calculus to approximate the value of a limit before confirming it with more precise methods. When approaching the study of limits, one powerful method is to first observe what happens to the function's values as they get closer to the point of interest—in this case, as \( x \) approaches 1. To do this, you can create a table of function values, choosing \( x \) values that get incrementally closer to 1, both from the left side (values like 0.9, 0.99, etc.) and from the right side (values like 1.1, 1.01, etc.).

• Evaluate the function \( 2x^2 + x - 4 \) for each chosen value of \( x \).
• Observe the pattern or trend in the results: Are they converging to a specific number?

By identifying the trend, you can estimate the limit of the function as \( x \) approaches 1. This practical approach helps build an intuitive understanding of the function's behavior around the point, providing a foundational step in your limit analysis.

Substitution method

In the context of finding limits, one of the simplest yet effective methods is direct substitution. This method involves plugging the value that \( x \) is approaching directly into the function. For our particular problem, as \( x \) approaches 1, you substitute \( x = 1 \) into the function \( 2x^2 + x - 4 \).
Here's why substitution is useful:

• It provides an immediate answer if the function is continuous at the point of interest, meaning the function doesn't have any holes or breaks there.
• When direct substitution results in a real number, like in this exercise where substitution yielded \(-1\), it confirms that the limit exists at that point.

However, remember that this method only applies when the function is uninterrupted. This isn't always the case, and that's when you might need to explore other techniques like factoring or rationalizing. In our problem, substitution provides a direct path to determining that \(-1\) is indeed the limit.

Simplifying expressions

Simplifying expressions is a crucial step in solving limit problems. This refers to reducing a mathematical expression to its simplest form, making it easier to evaluate or interpret. In our problem, after substituting \( x = 1 \) into the expression \( 2x^2 + x - 4 \), we proceed by simplifying.

• First, perform any necessary calculations. Here, calculate the exponent: \( 1^2 = 1 \).
• Following the order of operations (PEMDAS/BODMAS), carry out multiplications and divisions before addition and subtraction: Calculate \( 2 \times 1 = 2 \).
• Finally, add or subtract left-to-right: Perform \( 2 + 1 - 4 \), which results in \(-1 \).

Simplifying ensures the problem is in its most manageable form. This makes analyzing the limit more straightforward, and you avoid errors that can often occur with longer, more complex expressions. In our specific example, simplifying confirmed that the limit of the function as \( x \) approaches 1 is \(-1\).