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Magazine Chapter 2: Problem 39
Find the limit, if it exists. $$\lim _{x \rightarrow 3} \frac{x^{2}-9}{2 x-6}$$
Short Answer
Expert verified
The limit is 3.
Step by step solution
01
Identify the limit problem
The given limit problem is \(\lim _{x \rightarrow 3} \frac{x^{2}-9}{2 x-6}\). We need to find the value the expression approaches as \( x \) approaches 3.
02
Simplify the expression
First, simplify the expression \( \frac{x^2 - 9}{2x - 6} \). Notice that both the numerator and the denominator can be factored.
03
Factor the numerator
Factor the numerator \( x^2 - 9 \) as \( (x - 3)(x + 3) \). So, \ \frac{x^2 - 9}{2x - 6} \ becomes \ \frac{(x - 3)(x + 3)}{2x - 6} \.
04
Factor the denominator
Factor the denominator \( 2x - 6 \) as \( 2(x - 3) \). So, the expression now is \ \frac{(x-3)(x+3)}{2(x-3)} \.
05
Simplify the expression by canceling common factors
Cancel the common factor \( x - 3 \) in the numerator and denominator. This simplifies it to \ \frac{x + 3}{2} \.
06
Substitute the limit value
Now, substitute \( x = 3 \) into the simplified expression \ \frac{x + 3}{2} \. This gives \ \frac{3 + 3}{2} = \frac{6}{2} = 3 \.
07
State the result
Therefore, the limit is 3.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limit Evaluation
In calculus, evaluating a limit helps us understand the behavior of a function as the input approaches a certain value. When dealing with limits, we often have to simplify complex expressions to find the limit. For example, in the exercise, we need to determine the limit of \( \frac{x^2 - 9}{2x - 6} \) as \( x \) approaches 3. By following specific steps, we can find the value the function approaches when the input is very close to 3. These steps include simplifying the expression, often by factoring, and then carefully substituting the approaching value into the simplified form.
Factoring Polynomials
Factoring polynomials is crucial to simplifying expressions. In our example, the numerator \( x^2 - 9 \) is a difference of squares, which factors into \( (x - 3)(x + 3) \). Similarly, the denominator \( 2x - 6 \) factors to \( 2(x - 3) \). Recognizing these factorable forms is essential when simplifying, as it can reveal common factors in the numerator and denominator that can be canceled out. This step is key to reducing the expression to a simpler form that is easier to evaluate.
Simplifying Rational Expressions
After factoring, the next step is to simplify the rational expression. From the exercise, we initially have \( \frac{(x - 3)(x + 3)}{2(x - 3)} \). Notice the common factor \( x - 3 \) in both the numerator and the denominator. Canceling out these common factors simplifies the expression to \( \frac{x + 3}{2} \). Simplifying rational expressions by canceling common factors is a valuable technique because it removes the elements that complicate evaluation, leading us to a much simpler form of the function.
Substitution in Limits
Once the expression is simplified, substituting the approaching value is straightforward. From the previous step, we have \( \frac{x + 3}{2} \). We now substitute \( x = 3 \) to get \( \frac{3 + 3}{2} = \frac{6}{2} = 3 \). Substitution in limits involves replacing the variable with its approaching value in the simplified expression. This step verifies the limit and provides a concrete value that the function approaches as \( x \) gets close to the specified value.
