Problem 23 - University Calculus: Early Transcendentals

Question

Use I'Hópital's rule to find the limits. $$\lim _{t \rightarrow 0} \frac{t(1-\cos t)}{t-\sin t}$$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Check the Indeterminate Form

First, we need to determine if the limit presents an indeterminate form. By substituting \( t = 0 \) in the expression, we have:\[\frac{0 \cdot (1 - \cos 0)}{0 - \sin 0} = \frac{0}{0}.\]This is one of the indeterminate forms where L'Hôpital's Rule can be applied.

Step 2: Identify and Differentiate the Functions

To use L'Hôpital's Rule, differentiate the numerator and the denominator separately:The numerator is \( f(t) = t(1 - \cos t) \). Apply the product rule for differentiation:\[f'(t) = (1)(1 - \cos t) + t(\sin t).\]So, \( f'(t) = 1 - \cos t + t \sin t \).The denominator is \( g(t) = t - \sin t \). Differentiate to get:\[g'(t) = 1 - \cos t.\]

Step 3: Apply L'Hôpital's Rule

Since both the numerator and the denominator differentiated give us valid derivatives, apply L'Hôpital’s Rule:\[\lim _{t \rightarrow 0} \frac{f'(t)}{g'(t)} = \lim _{t \rightarrow 0} \frac{1 - \cos t + t \sin t}{1 - \cos t}.\]

Step 4: Simplify and Calculate the Limit

Simplify the expression obtained after applying L'Hôpital's Rule:Substitute \( t = 0 \) into the simplified limit:\[= \lim_{t \to 0} \left(1 + \frac{t \sin t}{1 - \cos t} \right).\]Insert \( t = 0 \) into this expression:\[= 1 + \lim_{t \to 0} \frac{0 \cdot \sin 0}{1 - \cos 0} = 1 + 0 = 1.\]

Step 5: Conclusion: Interpret the Result

The limit of the original expression as \( t \rightarrow 0 \) is equal to 1. Therefore, the given limit evaluates to 1 using L'Hôpital's Rule.

Key Concepts

Indeterminate Form

In calculus, an indeterminate form arises when substituting values into a mathematical expression provides a result that is not well-defined, like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). These forms indicate uncertainty and require further analysis to find the actual limit.
Understanding indeterminate forms is necessary for solving limit problems using L'Hôpital's Rule. This technique applies when substitution results in limited forms like \(\frac{0}{0}\).
Here's a quick reminder:

• Indeterminate forms include expressions like \(0/0\), \(\infty/\infty\), and \(0 \times \infty\).
• L'Hôpital's Rule can often resolve these by differentiating the numerator and denominator.
• Not all forms can be tackled with this rule, so understanding the type is crucial.

Differentiation

Differentiation is a key mathematical process used in calculus, which involves calculating a function's derivative. A derivative measures a function's rate of change as one of its variables changes.
L'Hôpital's Rule requires differentiating both the numerator and denominator of a rational function that yields an indeterminate form after substitution.
Differentiation has several useful techniques:

• Product Rule: Used when differentiating the product of two functions. If \(u(t)\) and \(v(t)\) are functions of \(t\), then \((uv)' = u'v + uv'\).
• Chain Rule: Utilized when a function is composed of other functions. For a composite function \(f(g(t))\), the derivative is \(f'(g(t)) \cdot g'(t)\).
• Quotient Rule: Applied to the division of two functions. For \(\frac{u}{v}\), the derivative is \(\frac{u'v - uv'}{v^2}\).

In the example provided, differentiation is critical in moving from an indeterminate form to a well-defined expression that allows finding the limit.

Limits

The concept of a limit is fundamental in calculus, helping to understand the behavior of a function as it approaches a specific point. Limits allow for the evaluation of functions at points where they are not necessarily explicitly defined.
Certain formulas, like \(\frac{0}{0}\), suggest the need for finding the limit to understand what's happening at specific points.
Key insights include:

• The limit of a function \(f(t)\) as \(t\) approaches a value \(a\) is the value that \(f(t)\) gets closer to as \(t\) gets closer to \(a\).
• Limits can handle points where the function does not initially have a value, as seen in applications of L'Hôpital's Rule.
• Finding limits is essential for calculating derivatives, integrals, and dealing with continuous functions.

In the given exercise, understanding limits was instrumental in simplifying the function to find that the expression, as \(t \ to 0\), results in the value of 1.