Problem 367 - Calculus Volume 1

Question

For the following exercises, evaluate the limits with either L'Hôpital's rule or previously learned methods. $$ \lim _{x \rightarrow 3} \frac{x^{2}-9}{x-3} $$

Step-by-Step Solution

Answer

The limit is 6.

Step 1: Identify the Indeterminate Form

The limit expression given is \( \frac{x^2 - 9}{x - 3} \). First, substitute \( x = 3 \) into the expression to check for an indeterminate form. Substituting gives \( \frac{3^2 - 9}{3 - 3} = \frac{0}{0} \), which is an indeterminate form. This means we can use L'Hôpital's rule or other algebraic simplification methods to evaluate the limit.

Step 2: Factor the Numerator

Notice that the numerator \( x^2 - 9 \) is a difference of squares, which can be factored as \( (x-3)(x+3) \). This gives us a new expression: \( \frac{(x-3)(x+3)}{x-3} \).

Step 3: Simplify the Expression

Now, simplify the expression \( \frac{(x-3)(x+3)}{x-3} \) by canceling the \( x-3 \) terms from the numerator and the denominator, yielding the simpler expression \( x+3 \).

Step 4: Evaluate the Limit of the Simplified Expression

Substitute \( x = 3 \) into the simplified expression \( x+3 \). This gives \( 3+3 = 6 \). Therefore, the limit is \( 6 \).

Key Concepts

Understanding Indeterminate Forms

In calculus, indeterminate forms occur when substituting a value into a limit expression results in an undefined form. These forms often appear as \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), and others that are ambiguous. In the context of limits, these forms indicate that direct substitution does not provide enough information about the limit's behavior. L'Hôpital's Rule is a powerful tool for resolving indeterminate forms like \( \frac{0}{0} \).
By recognizing indeterminate forms, we can apply appropriate techniques, such as L'Hôpital's Rule or algebraic manipulations, to evaluate the limit. In our example, substituting \( x = 3 \) leads to \( \frac{0}{0} \), a classic indeterminate form that signals further work is needed to find the limit.

Factoring Algebraic Expressions

Factoring is a strategy used to simplify expressions, often revealing cancellations that aren't immediately obvious. In our exercise, the expression \( x^2 - 9 \) is a difference of squares.
This means it can be factored into \((x-3)(x+3)\).
Factoring helps in identifying cancellations that simplify the expression for limit calculation. It's important to apply factoring techniques correctly:

• Notice common patterns like difference of squares: \( a^2 - b^2 = (a-b)(a+b) \)
• Factor the expression completely, if possible
• Look for terms that can be canceled to simplify the expression

In this case, factoring allows us to cancel \( x-3 \) in both the numerator and denominator, simplifying the expression to \( x+3 \). This step drastically simplifies the evaluation process.

Evaluating Limits Effectively

Evaluating limits involves finding the value that a function approaches as the input approaches a specific point. Once the expression is simplified, as in our example where it became \( x + 3 \), substituting the desired limit value becomes straightforward.
To evaluate limits effectively:

• Always check the expression for indeterminate forms first
• Use algebraic simplifications like factoring to cancel ambiguous terms
• Substitute the limit value into the simplified expression for a clear result

In this exercise, simplifying to \( x+3 \) and substituting \( x = 3 \) gives us \( 3 + 3 = 6 \). Thus, the limit of the original expression is 6. This method shows how breaking down the problem into manageable steps can clarify complex exercises.