Problem 18 - Thomas Calculus

Question

Use 1'Hopital's Rule to find the limits in Exercises \(7-26\). $$ \lim _{t \rightarrow 0} \frac{10(\sin t-t)}{t^{3}} $$

Step-by-Step Solution

Answer

The limit is \(-\frac{5}{3}\).

Step 1: Identify the Indeterminate Form

First, substitute \( t = 0 \) into the expression \( \frac{10(\sin t - t)}{t^3} \). This yields \( \frac{10(0 - 0)}{0^3} = \frac{0}{0} \). Since this is an indeterminate form \( \frac{0}{0} \), we can apply L'Hopital's Rule.

Step 2: Apply L'Hopital's Rule

According to L'Hopital's Rule, for functions \( f(t) \) and \( g(t) \) where \( \lim_{t \to a} \frac{f(t)}{g(t)} = \frac{0}{0} \), we have that \( \lim_{t \to a} \frac{f(t)}{g(t)} = \lim_{t \to a} \frac{f'(t)}{g'(t)} \) if this limit exists. Here, take the derivative of the numerator \( 10(\sin t - t) \) and the denominator \( t^3 \).

Step 3: Differentiate the Numerator and Denominator

Differentiate the numerator: \( f(t) = 10(\sin t - t) \). The derivative is \( f'(t) = 10(\cos t - 1) \).Differentiate the denominator: \( g(t) = t^3 \). The derivative is \( g'(t) = 3t^2 \).

Step 4: Apply L'Hopital's Rule Again

Substituting back into the limit, we have: \( \lim_{t \to 0} \frac{10(\cos t - 1)}{3t^2} \). Substituting \( t = 0 \) results in \( \frac{10(1 - 1)}{0} = \frac{0}{0} \) again. We need to apply L'Hopital's Rule one more time.

Step 5: Differentiate Again

Differentiate the new numerator: \( f'(t) = 10(\cos t - 1) \). The derivative is \( f''(t) = -10 \sin t \).Differentiate the new denominator: \( g'(t) = 3t^2 \). The derivative is \( g''(t) = 6t \).

Step 6: Evaluate the New Limit

Substitute back into the limit:\( \lim_{t \to 0} \frac{-10 \sin t}{6t} \). Substitute \( t = 0 \) into the expression, resulting in \( \frac{-10 \cdot 0}{6 \cdot 0} = \frac{0}{0} \). Again the form is indeterminate, so apply L'Hopital's Rule one last time.

Step 7: Differentiate Final Time

Differentiate the final numerator: \( -10 \sin t \). The derivative is \( -10 \cos t \).Differentiate the final denominator: \( 6t \). The derivative is \( 6 \).

Step 8: Solve the Limit After Final Differentiation

Substitute in the final derivatives: \( \lim_{t \to 0} \frac{-10 \cos t}{6} = \frac{-10 \cdot 1}{6} = \frac{-10}{6} = -\frac{5}{3} \). The limit exists and evaluates to \(-\frac{5}{3}\).

Key Concepts

Indeterminate Forms

Indeterminate forms are expressions in calculus where substitution results in undefined or ambiguous terms, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). They indicate that simpler direct evaluation of limits won't work.
When you encounter an indeterminate form, it tells you that more work is needed to find the limit properly. Types of Indeterminate Forms Include:

• \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \)
• \( 0 \cdot \infty \)
• \( \infty - \infty \)
• \( 0^0, \infty^0, \) and \( 1^\infty \)

In our exercise, we have the \( \frac{0}{0} \) form, prompting the use of L'Hopital's Rule.
Remember, identifying this form helps decide the right strategy to solve the limit.

Limit Evaluation

Limit evaluation is about finding the value that a function approaches as the input approaches some point.
This is a foundational concept in calculus and helps us understand the behavior of functions near specific points. Methods for Evaluating Limits Include:

• Direct Substitution: Use when plugging in the value leads to a defined result.
• Factoring: Simplify the expression by canceling out common terms.
• Conjugate Multiplication: Use to rationalize expressions with radicals.
• L'Hopital's Rule: A special technique for dealing with indeterminate forms.

In the given problem, we use L'Hopital's Rule due to encountering an indeterminate form. Evaluating the limit correctly is crucial for understanding how the function behaves at \( t = 0 \).
The goal is to converge on a finite value or confirm divergence.

Differentiation

Differentiation is the process of finding the derivative, which represents the rate of change of a function.
It is a key tool in calculus often needed for applying L'Hopital's Rule.
Key Points about Differentiation:

• The derivative of a function \( f(t) \) is noted as \( f'(t) \).
• Basic rules include the power rule, product rule, quotient rule, and chain rule.
• For trigonometric functions, \( \sin \) and \( \cos \), the derivatives are \( \cos(t) \) and \( -\sin(t) \) respectively.

In our example, differentiating both the numerator and the denominator was critical to applying L'Hopital's Rule.
Remember to differentiate multiple times if needed, as in this exercise, until a determinate form is reached.

Trigonometric Limits

Trigonometric limits often require special attention, as they involve the behavior of sine, cosine, and tangent functions near specific points, usually \( 0 \), \( \pi \), or \( \frac{\pi}{2} \).
These functions have unique properties and periodic characteristics which help in evaluating limits. Common Trigonometric Limit Properties:

• \( \lim_{t \to 0} \frac{\sin t}{t} = 1 \)
• \( \lim_{t \to 0} \frac{1 - \cos t}{t} = 0 \)
• Symmetry and periodicity play a vital role in simplifying limits.

Using these properties, we handle more complex trigonometric limits effectively.
In our scenario, it was necessary to differentiate twice due to its indeterminate nature. Understanding these basic limits and properties simplifies solving complex expressions.