Problem 14 - Calculus for Biology and Medicine

Question

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 2} \frac{x^{2}-4}{x+2} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understanding the Expression

We are tasked with finding the limit \( \lim _{x \rightarrow 2} \frac{x^{2}-4}{x+2} \). This expression can be simplified. The numerator is a difference of squares, which means it can be factored.

Step 2: Factoring the Numerator

The numerator \( x^2 - 4 \) factors into \( (x-2)(x+2) \). Thus, the expression becomes \( \frac{(x-2)(x+2)}{x+2} \).

Step 3: Simplifying the Expression

Simplify the expression \( \frac{(x-2)(x+2)}{x+2} \) by canceling out the \( x+2 \) terms in the numerator and denominator. This is valid for all \( x eq -2 \). The simplified expression is \( x-2 \).

Step 4: Evaluating the Limit

To find the limit as \( x \rightarrow 2 \), we substitute \( x = 2 \) into the simplified expression \( x-2 \). This yields \( 2-2 = 0 \).

Step 5: Verification Using Numerical Table

To ensure accuracy, create a table with values approaching \( x = 2 \) from both sides (e.g., 1.9, 1.99, 2.01, 2.1). When substituting these values into \( x-2 \), the results should approximate 0, confirming our solution.

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential algebraic technique, especially when dealing with expressions that can be simplified. In the expression \( x^2 - 4 \), we see an opportunity to factor a perfect square difference. This form, known as the difference of squares, is expressed as \( a^2 - b^2 \).

Here's a quick breakdown:

• The difference of squares can always be factored into \( (a-b)(a+b) \).
• For \( x^2 - 4 \), we identify \( a = x \) and \( b = 2 \), so we write it as \((x-2)(x+2)\).
• Factoring helps in simplifying limits and can be crucial in solving calculus problems more systematically.

By rewriting \( x^2-4 \) as \( (x-2)(x+2) \), we have a clearer view of the expression, which simplifies the limit problem we're dealing with.

Simplifying Expressions

Simplifying an expression means reducing it to its simplest form. This process involves canceling common factors between the numerator and the denominator, as facilitated by factoring.
In our limit problem, upon factoring \( x^2-4 \) to \( (x-2)(x+2) \), the fraction \( \frac{(x-2)(x+2)}{x+2} \) allows for further simplification.

• When we cancel the \( x+2 \) in the numerator and denominator, we need to keep in mind that this step is only valid when \( x eq -2 \).
• The result of the simplification, \( x-2 \), is much easier to handle and compute as \( x \) approaches 2.

This simplification makes it straightforward to evaluate the limit, and underscores the utility of factoring as a tool in calculus.

Numerical Methods

Numerical methods involve using numerical approximations to gain insights into mathematical problems. In limit problems, especially when algebraic simplification is not possible, numerical methods provide an excellent alternative.
In our exercise, after simplifying to \( x-2 \), employing a numerical table helps ensure the solution is accurate.

• Create a table with values approaching \( x = 2 \) from both sides, such as 1.9, 1.99, 2.01, and 2.1.
• Substitute these values into the simplified expression \( x-2 \) to observe the output.
• As these values approach 2, the expression \( x-2 \) approaches 0.

Using a numerical method, like constructing a table, helps to confirm the solution and provides insight into the behavior of functions at certain points.