Problem 11 - Essential Calculus Early Transcendentals

Question

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{t \rightarrow 1} \frac{t^{8}-1}{t^{5}-1}$$

Step-by-Step Solution

Answer

The limit is \(\frac{8}{5}\).

Step 1: Identify the Form

We need to determine if we have an indeterminate form. Substituting \(t=1\) gives us \(\frac{1^8-1}{1^5-1}=\frac{0}{0}\). This is an indeterminate form.

Step 2: Apply l'Hospital's Rule

Since we have an indeterminate form \(\frac{0}{0}\), we can apply l'Hospital's Rule. This means we will take the derivative of the numerator and the denominator.

Step 3: Differentiate the Numerator and Denominator

The derivative of the numerator \(t^8 - 1\) is \(8t^7\). The derivative of the denominator \(t^5 - 1\) is \(5t^4\). So, using l'Hospital's Rule, we consider the limit: \[\lim_{t \to 1} \frac{8t^7}{5t^4}.\]

Step 4: Simplify and Evaluate the New Limit

Now simplify \(\frac{8t^7}{5t^4}\) by reducing: \(\frac{8t^7}{5t^4} = \frac{8t^{7-4}}{5} = \frac{8t^3}{5}\). Now evaluate the limit as \(t\) approaches 1: \[\lim_{t \to 1} \frac{8t^3}{5} = \frac{8(1)^3}{5} = \frac{8}{5}.\]

Key Concepts

Indeterminate Form

When evaluating limits, an important thing to check for is whether the expression results in an indeterminate form. This occurs when substituting the value into the limit gives an unclear result like \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), or \(0 \cdot \infty\). These forms do not provide sufficient information about the limit's behavior, making them appear ambiguous.

In our exercise, substituting \(t = 1\) into \(\lim_{t \to 1} \frac{t^8 - 1}{t^5 - 1}\) yields \(\frac{0}{0}\), indicating the presence of an indeterminate form. This tells us that further analysis or a special method, like l'Hospital's Rule, is needed to evaluate the limit successfully.

Identifying indeterminate forms is crucial because they signal that direct substitution isn't sufficient and a deeper investigation is required.

Limit Evaluation

Evaluating limits involves determining how a function behaves as it approaches a specific point. In many cases, direct substitution suffices. However, when indeterminate forms are present, additional techniques must be employed.

One common approach is to use algebraic simplification, but when that isn't practical or effective, advanced methods like l'Hospital's Rule come into play. For the exercise provided, directly evaluating \(\lim_{t \to 1} \frac{t^8 - 1}{t^5 - 1}\) resulted in an indeterminate form, signaling the need for l'Hospital's Rule.

This step emphasizes the need for flexibility in problem-solving. Not every limit can be solved as presented; understanding the conditions and techniques applicable is essential for finding definitive answers.

Derivatives

Derivatives play a vital role in l'Hospital's Rule. They allow us to re-evaluate the limit by differentiating the numerator and the denominator of the original fraction.

In the provided solution, the function \(\frac{t^8 - 1}{t^5 - 1}\) was differentiated as follows:

• The derivative of the numerator \(t^8 - 1\) is \(8t^7\).
• The derivative of the denominator \(t^5 - 1\) is \(5t^4\).

Applying these derivatives to the expression, we consider \(\lim_{t \to 1} \frac{8t^7}{5t^4}\). This new expression is simpler and free of the indeterminate form, thus allowing us to continue with our evaluation.

Understanding derivatives' role in limit evaluation helps solve problems involving complex expressions and is a fundamental concept in calculus.

Simplifying Expressions

Once derivatives are calculated, the next step involves simplifying the expression as much as possible to make evaluation straightforward. Such simplification can reveal the value of the limit.

In our exercise, after applying l'Hospital's Rule, the expression \(\frac{8t^7}{5t^4}\) was simplified to \(\frac{8t^3}{5}\). This was achieved by reducing the powers of \(t\), confirming that the limit evaluates to \(\frac{8}{5}\) as \(t\) approaches 1.

Simplifying expressions correctly can transform a formerly complex problem into one that is easily manageable. This highlights the importance of algebraic manipulation in calculus, underscoring how a systematic approach can lead to clarity and resolution.