Problem 31 - Precalculus

Question

Use a table of values to evaluate each function as \(x\) approaches the value indicated. If the function seems to approach a limiting value, write the relationship in words and using the limit notation. $$f(x)=\frac{x^{2}-2 x}{x^{2}-4} ; x \rightarrow 2$$

Step-by-Step Solution

Answer

The limit of \( f(x) \) as \( x \to 2 \) is 1: \( \lim_{{x \to 2}} f(x) = 1 \).

Step 1: Identify the function and approach value

The given function is \( f(x) = \frac{x^2 - 2x}{x^2 - 4} \), and we need to evaluate the limit as \( x \to 2 \). The expression \( x^2 - 4 \) can be factored to \((x - 2)(x + 2)\).

Step 2: Simplify the expression

Before creating a table of values, simplify the function if possible. The function \( f(x) \) can be rewritten as \[ \frac{x(x - 2)}{(x - 2)(x + 2)} \]. Notice that \( (x - 2) \) can be canceled from the numerator and the denominator for \( x eq 2 \), simplifying it to \[ \frac{x}{x + 2} \].

Step 3: Create a table of values

Choose values of \( x \) close to 2, both from the left and right. Calculate \( f(x) \) for each value:| \( x \) | \( f(x) \) ||-------|-------------|| 1.9 | 0.987 || 1.99 | 0.9987 || 1.999 | 0.99987 || 2.001 | 1.00013 || 2.01 | 1.0013 || 2.1 | 1.013 |

Step 4: Observe the values

As \( x \) approaches 2 from both sides, the values of \( f(x) \) get closer to 1. Hence, we can infer that the function seems to approach a limiting value as \( x \to 2 \).

Step 5: Write the relationship in words and limit notation

In words, the value of the function \( f(x) \) approaches 1 as \( x \) gets closer to 2. In limit notation, this is written as \[ \lim_{{x \to 2}} f(x) = 1 \].

Key Concepts

function simplification

Function simplification is a key step in solving limit problems because it helps reduce the complexity of the function before evaluating its limit.

• Identify components of the function that can be factored or canceled.
• Factors common to the numerator and the denominator can be canceled when evaluating a limit, to simplify the expression.

For instance, in the function \( f(x) = \frac{x^2 - 2x}{x^2 - 4} \), \( x^2 - 4 \) can be factored to \((x - 2)(x + 2)\). This simplification allows the cancellation of \( (x-2) \) in both the numerator and denominator. Be mindful that this cancellation is only valid for \(x eq 2\), because at \(x = 2\) the factor would make both the numerator and denominator zero. After the simplification, the function becomes \( \frac{x}{x+2} \) which is much easier to evaluate near \( x = 2 \). Simplification helps avert undefined terms and simplifies calculations, vital for understanding the behavior of the function as the limit approaches.

limit evaluation

Limit evaluation is about understanding the behavior of a function as the variable approaches a certain point. Once a function is simplified, you can more easily evaluate its limit by considering values close to the given point.

• Substitute values closer to the point from both the left and right.
• Observe the trend or pattern in the function values as the input nears the target.

In the example, after simplifying the function to \( \frac{x}{x+2} \), you would evaluate the limit as \( x \) approaches 2. By substituting values like 1.99, 1.999, 2.001, and so on, you can calculate corresponding \( f(x) \) values.

• For \( x = 1.9 \), \( f(x) \approx 0.987 \)
• For \( x = 2.1 \), \( f(x) \approx 1.013 \)
• Observe that as \( x \) closes in on 2, \( f(x) \) approaches 1.

This consistent behavior implies that the function tends to 1, fully understanding the trend expects knowing how values veering towards the point influence the function.

limit notation

Limit notation is a concise way of expressing the trend that a function demonstrates as its input approaches a particular value. Crucial in communicating mathematical limits, it shows the approximate behavior of functions around specific points.The standard format for limit notation is \( \lim_{{x \to c}} f(x) = L \), where:

• \(x \to c\) denotes \(x\) approaching a specific value \(c\).
• \(f(x)\) is the function being evaluated.
• \(L\) represents the resulting limit value.

For the exercise given, as \( x \to 2 \), \( f(x) = \frac{x}{x+2} \) approaches 1. It tells us that the function values are getting indefinitely close to 1 when \( x \) gets closer to 2. Thus, this written in limit notation is: \( \lim_{{x \to 2}} f(x) = 1 \).Limit notation succinctly informs mathematicians and students of the behavior of a function without computing infinite values, encompassing the essence of continuity and change in calculus.