Problem 7 - University Calculus: Early Transcendentals

Question

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4}$$

Step-by-Step Solution

Answer

The limit is \( \frac{1}{4} \).

Step 1: Recognize the Form

First, identify if the limit problem is in an indeterminate form, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Substitute \( x = 2 \) into the numerator and the denominator: \( x - 2 = 0 \) and \( x^2 - 4 = 0 \). Therefore, we have a \( \frac{0}{0} \) indeterminate form.

Step 2: Apply L'Hôpital's Rule

Since the limit is in a \( \frac{0}{0} \) form, we can apply L'Hôpital's Rule, which states that for limits of the form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), the limit of \( \frac{f(x)}{g(x)} \) as \( x \to a \) is the same as \( \frac{f'(x)}{g'(x)} \) as \( x \to a \), provided the limit on the right exists.

Step 3: Differentiate the Numerator and Denominator

Differentiate both the numerator \( x - 2 \) and the denominator \( x^2 - 4 \). The derivative of the numerator \( x - 2 \) is \( 1 \). The derivative of the denominator \( x^2 - 4 \) is \( 2x \).

Step 4: Re-evaluate the Limit

Now, we can evaluate the limit of the derivatives: \[ \lim_{x \to 2} \frac{1}{2x} \]. Substitute \( x = 2 \) into \( \frac{1}{2x} \) to get \( \frac{1}{4} \).

Step 5: State the Final Answer

The limit of \( \frac{x-2}{x^{2}-4} \) as \( x \to 2 \) is \( \frac{1}{4} \).

Key Concepts

Indeterminate Forms

When solving limit problems, particularly with L'Hôpital's Rule, the first step is identifying the type of limit problem you are dealing with. Indeterminate forms are a special category where straightforward evaluation does not lead to a clear result. In the context of limits, the most common indeterminate forms are \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \). These forms signal that more work is needed to find the actual limit.Imagine you are evaluating \( \lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4}\). By substituting \(x = 2\), you find both the numerator and the denominator turn to zero, resulting in the form \( \frac{0}{0} \). This triggers the possible usage of L'Hôpital's Rule.This rule is particularly helpful for such situations because it allows you to transform the original problem into something easier to evaluate. Remember:

• Check if the limit leads to an indeterminate form first.
• L'Hôpital's Rule can only apply if your problem fits into these specific forms.

Derivatives

Understanding derivatives is crucial in applying L'Hôpital's Rule. A derivative essentially measures how a function changes as its input changes. In practical terms, the derivative provides the slope of the tangent line to the graph of the function at any point.In using L'Hôpital's Rule, once you've identified an indeterminate form, you must differentiate both the numerator and the denominator independently. This involves finding the derivative of each function separately.For the example given, where the expression is \( \frac{x-2}{x^{2}-4} \), the derivatives are:

• Numerator \( x - 2 \): derivative is \( 1 \).
• Denominator \( x^2 - 4 \): derivative is \( 2x \).

When you differentiate correctly, the problem transitions from being unsolvable by direct substitution to something that is much clearer by re-evaluating \( \frac{1}{2x} \).These derivatives transform the original limit into a simpler form that can be directly evaluated.

Limit Evaluation

After finding the derivatives, it's time to re-evaluate the limit using a simpler expression. This is where the power of L'Hôpital's Rule becomes apparent and the complexity of indeterminate forms is resolved.Initially, it was difficult to evaluate \( \lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4} \) directly due to the indeterminate form \( \frac{0}{0} \). However, once differentiated, the limit transforms into \( \lim_{x \to 2} \frac{1}{2x} \). This expression is easy to evaluate by direct substitution.Simply plug \( x = 2 \) into the differentiated expression. This gives you \( \frac{1}{2 \times 2} = \frac{1}{4} \). Thus, the original limit, \( \lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4} \), resolves to \( \frac{1}{4} \).Remember when evaluating limits:

• Always check if L'Hôpital's Rule applies.
• Simplify the derivatives before re-evaluating.
• Substitute values directly once the expression is simplified.