Problem 11 - Calculus Early Transcendentals

Question

Let \(f(x)=\frac{x^{2}-4}{x-2}\) a. Calculate \(f(x)\) for each value of \(x\) in the following table. b. Make a conjecture about the value of \(\lim _{x \rightarrow 2} \frac{x^{2}-4}{x-2}\)

Step-by-Step Solution

Answer

Answer: The conjecture for the limit of the function \(f(x) = \frac{x^2 - 4}{x - 2}\) as \(x\) approaches 2 is 4.

Step 1: 1. Function Evaluation:

To evaluate the function at the given \(x\) values, follow these steps: 1. Substitute the value of the \(x\) into \(f(x)\). 2. Simplify the expression using rules of algebra.

Step 2: 2. Function Evaluation Example:

Let's evaluate \(f(x)\) at an example value of \(x=1\): 1. Substitute the value of \(x\) into \(f(x)\): \(f(1)=\frac{1^{2}-4}{1-2}\) 2. Simplify the expression using algebra: \(f(1)=\frac{-3}{-1}=3\) Repeat the same process for each \(x\) value in the table. **Part b - Conjecture the limit of \(f(x)\) as \(x\) approaches 2**

Step 3: 1. Observing Function Evolution:

Look at how the function \(f(x)\) behaves as \(x\) gets increasingly close to 2 from both the left and right side. Make sure to avoid directly plugging 2 into the function, as this would result in a division by zero.

Step 4: 2. Factoring the Function:

An alternative approach is to simplify the function algebraically to remove the possibility of division by zero at \(x=2\). Consider factoring the numerator of \(f(x)\): $$\frac{x^{2}-4}{x-2}=\frac{(x-2)(x+2)}{x-2}$$

Step 5: 3. Simplifying the Function and Taking the Limit:

We can now cancel out the \((x-2)\) terms in the numerator and denominator, which were causing the division by zero: $$\frac{(x-2)(x+2)}{x-2}=x+2$$ Now, we can find the limit as \(x\) approaches 2: $$\lim_{x \rightarrow 2}(\frac{x^{2}-4}{x-2})=\lim_{x \rightarrow 2}(x+2)=2+2=4$$

Step 6: 4. Conjecture:

The conjecture is that the value of the limit of \(f(x)\) as \(x\) approaches 2 is 4: $$\lim _{x \rightarrow 2} \frac{x^{2}-4}{x-2}=4$$

Key Concepts

Function Evaluation

Understanding how to evaluate a function is a fundamental skill in mathematics. When we talk about function evaluation, we refer to the process of determining the output of a function for a specific input. For the function \( f(x) = \frac{x^2 - 4}{x - 2} \), to evaluate \( f(x) \) for different values of \( x \), we substitute these values into the function and simplify.

• Substitution: Replace the variable \( x \) with the given number.
• Simplification: Use algebraic rules such as combining like terms, factoring, and cancelling to simplify the expression.

For example, evaluating \( f(x) \) when \( x = 1 \), we substitute and get \( f(1) = \frac{1^2 - 4}{1 - 2} = \frac{-3}{-1} = 3 \). If you do this for a range of values and observe the function's behavior, it helps in understanding how the function behaves as the input gets close to a certain value, particularly when that value results in an undefined expression.

Algebraic Simplification

Algebraic simplification is the process of making an algebraic expression easier to work with by reducing it to its simplest form. This often involves factoring, expanding, combining like terms, and cancelling common factors. In the context of limits, algebraic simplification plays a crucial role when direct substitution leads to an undefined expression, such as division by zero.
Take the problem \( \frac{x^2 - 4}{x - 2} \) for example. We can factor the numerator as \( (x - 2)(x + 2) \), which reveals a common factor with the denominator. The factored form is \( \frac{(x - 2)(x + 2)}{x - 2} \). You can then cancel the \( (x - 2) \) term from the numerator and denominator, leaving us with the simplified function \( x + 2 \). This simplified version is much easier to work with, especially when calculating limits, as it allows for direct substitution without creating the problem of division by zero at \( x = 2 \).

Limits and Continuity

The concept of limits is foundational in calculus and deals with the behavior of functions as they approach a particular point. In our example, we are concerned with the limit of the function \( f(x) = \frac{x^2 - 4}{x - 2} \) as \( x \) approaches 2. Due to the division by zero, we cannot simply substitute 2 into the function. Instead, we must find a way to understand the behavior of the function near that point.
After algebraic simplification, as shown above, we find that the function can be expressed as \( x + 2 \) when \( x \) is not equal to 2. This simplified expression is continuous, meaning it does not have any breaks or holes. Therefore, we can now evaluate the limit by direct substitution. This gives us \( \lim\textsubscript{{x \rightarrow 2}} (\frac{x^2 - 4}{x - 2}) = \lim\textsubscript{{x \rightarrow 2}} (x + 2) = 2 + 2 = 4 \). Continuity is a related concept, indicating that the function has a connected and unbroken graph. For the limit of a function at a point to exist, the function must approach the same value from both left and right as it approaches that point, signaling the function's continuity at that point.