Problem 9 - University Calculus: Early Transcendentals

Question

Use I'Hópital's rule to find the limits. $$\lim _{r \rightarrow-3} \frac{t^{3}-4 t+15}{t^{2}-t-12}$$

Step-by-Step Solution

Answer

The limit is \(-\frac{23}{7}\).

Step 1: Identify the Indeterminate Form

To apply L'Hôpital's Rule, we must first confirm that the limit is in an indeterminate form. Evaluate the numerator and denominator separately by substituting \( t = -3 \). Numerator: \((-3)^3 - 4(-3) + 15 = -27 + 12 + 15 = 0\)Denominator: \((-3)^2 - (-3) - 12 = 9 + 3 - 12 = 0\)Since both the numerator and denominator equal zero, the limit is in the indeterminate form \( \frac{0}{0} \).

Step 2: Apply L'Hôpital's Rule

Since the limit is in an indeterminate form, we can use L'Hôpital's Rule, which involves taking the derivative of the numerator and the denominator separately, then reevaluating the limit.Let's differentiate the numerator: \( \frac{d}{dt}(t^3 - 4t + 15) = 3t^2 - 4 \)Now, differentiate the denominator: \( \frac{d}{dt}(t^2 - t - 12) = 2t - 1 \)

Step 3: Evaluate the New Limit

Substitute \( t = -3 \) into the derivatives obtained in Step 2.Numerator's derivative: \( 3(-3)^2 - 4 = 27 - 4 = 23 \)Denominator's derivative: \( 2(-3) - 1 = -6 - 1 = -7 \)The limit now becomes:\[ \lim_{t \to -3} \frac{23}{-7} = \frac{23}{-7} \]

Step 4: State the Final Result

Since applying L'Hôpital's Rule resolves the indeterminate form and the limit \( \frac{23}{-7} \) is a finite number, this is the solution to the original limit problem when \( t \to -3 \).

Key Concepts

Indeterminate Forms

When dealing with limits in calculus, you might encounter expressions that don't lead directly to a solution. These expressions often fall under what we call "indeterminate forms." The most common types of indeterminate forms are \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \). These forms are "indeterminate" because they don't provide sufficient information to deduce a specific numeric value right away.

For example, when you substitute \( t = -3 \) into the original limit problem, both the numerator and the denominator turn into 0. This is a classic case of the \( \frac{0}{0} \) indeterminate form, which tells us that we can’t determine the limit by direct substitution. Indeterminate forms highlight the need for more advanced techniques, like L'Hôpital's Rule, to evaluate the limit further.

Calculus Limits

At its core, the concept of a limit in calculus is about understanding what value a function approaches as the input approaches some value. Limits are foundational in calculus because they help us define other critical concepts, such as derivatives and integrals.

When dealing with limits, especially when functions are complex, the direct substitution of values into functions gives a valuable starting point. But sometimes, it results in an indeterminate form, signaling that the destination is not straightforward. This can occur when working with polynomials, rational functions, or those involving radicals or trigonometric functions.

The ultimate goal is to find a unique and finite number to which the function is tending, even if the substitution initially didn’t provide clarity. L'Hôpital's Rule becomes a handy tool in these cases.

Differentiation Techniques

Differentiation is a core technique in calculus that involves finding the derivative of a function. This derivative represents the rate of change of the function concerning its variable.

In L'Hôpital's Rule, we take advantage of differentiation to resolve indeterminate forms. Specifically, we differentiate the numerator and the denominator of a given fraction separately. For instance, given a limit problem with a \( \frac{0}{0} \) form, like our exercise, the rule states that we should find the derivatives of both the numerator and denominator.

• Numerator Derivative: Transform the function in the numerator to reflect its rate of change.
• Denominator Derivative: Similarly, find the rate of change of the denominator.

After differentiating, we attempt substitution again. The new function, ideally, will be a standard form that provides a clear answer. This technique is particularly powerful as it simplifies complex expressions and brings clarity where, initially, uncertainty reigned.