Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{t \rightarrow 0} \frac{1-\cot t}{1 / t} $$

Step-by-Step Solution

Answer

The limit does not exist due to division by zero and repeated indeterminate forms, indicating complexity around \( t = 0 \).

Step 1: Understanding the Limit Expression

The given limit expression is \( \lim _{t \rightarrow 0} \frac{1-\cot t}{1/t} \). The goal is to find this limit as \( t \) approaches 0.

Step 2: Simplifying the Expression

Observe the function \( \frac{1-\cot t}{1/t} \). Rewrite \( \cot t \) as \( \frac{\cos t}{\sin t} \). Thus, the expression becomes \( \frac{1 - \frac{\cos t}{\sin t}}{1/t} = \frac{\sin t - \cos t}{t \sin t} \).

Step 3: Expression Factorization and Substitution

Notice that \( \sin t \) in the denominator will cause the expression to be undefined at \( t = 0 \). Therefore, apply L'Hôpital's Rule which states that \( \lim_{t \to a} \frac{f(t)}{g(t)} = \frac{f'(t)}{g'(t)} \) if \( \lim_{t \to a} f(t) = \lim_{t \to a} g(t) = 0 \). First, find the derivative of the numerator and denominator.

Step 4: Derivative Calculation

Differentiate the numerator \( \sin t - \cos t \) to get \( \cos t + \sin t \). Differentiate the denominator \( t \sin t \) giving \( \sin t + t \cos t \).

Step 5: Applying L'Hôpital's Rule

The new expression after applying L'Hôpital's Rule is \( \frac{\cos t + \sin t}{\sin t + t \cos t} \). Evaluate this new limit as \( t \) approaches 0.

Step 6: Calculating the Limit

Substitute \( t = 0 \) into \( \frac{\cos t + \sin t}{\sin t + t\cos t} \). This reduces to \( \frac{\cos 0 + \sin 0}{\sin 0 + 0 \cdot \cos 0} = \frac{1 + 0}{0 + 0} \), which results in an indeterminate form. Apply L'Hôpital's Rule again if needed.

Step 7: Further Simplification for Evaluation

After applying L'Hôpital's Rule a second time, evaluate the expression numerically or graphically near \( t = 0 \) using a calculator.

Step 8: Verifying Limit with Graphing Calculator

Use a graphing calculator to plot the function \( \frac{\sin t - \cos t}{t \sin t} \) near \( t = 0 \), and observe the behavior as \( t \to 0 \). Confirm the result from your calculations.

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a mathematical tool that helps when evaluating limits that result in indeterminate forms such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). This rule states that if both the numerator \( f(t) \) and the denominator \( g(t) \) approach 0 or infinity as \( t \to a \), then:

• \( \lim_{t \to a} \frac{f(t)}{g(t)} = \lim_{t \to a} \frac{f'(t)}{g'(t)} \)
• as long as \( \lim_{t \to a} \frac{f'(t)}{g'(t)} \) exists or the limit approaches a finite value.

This makes it easier to evaluate complex limit expressions by differentiating the numerator and the denominator separately.
In the exercise, the rule was applied to the function \( \frac{1-\cot t}{1/t} \). Through simplification and differentiation, the limit could be evaluated even at points that initially led to undefined forms.

Trigonometric Limits

Trigonometric limits often involve more intricate algebraic manipulation. For trigonometric functions, limits analysis helps illustrate behavior as an angle approaches a particular value.
A useful technique includes recognizing standard trigonometric identities and potential simplifications. With the exercise's limit formula \( \frac{1-\cot t}{1/t} \), transforming \( \cot t \) into familiar trigonometric functions \( \cos t \) and \( \sin t \) was essential.

• Rewriting \( \cot t \) as \( \frac{\cos t}{\sin t} \) converts the expression into a more workable form \( \frac{\sin t - \cos t}{t \sin t} \).
• This conversion points to specific trigonometric limits, enabling further analysis through differentiation.

When evaluating limits at an indeterminate point, understanding trigonometric identities is crucial in seeking a path to simplification.

Graphing Calculator Techniques

Graphing calculators are powerful tools in visualizing mathematical functions. They help confirm algebraic calculations by providing a graphical representation.
For limits, using a calculator involves:

• Entering the function \( \frac{\sin t - \cos t}{t \sin t} \) into the calculator.
• Setting an appropriate viewing window around the limit point \( t = 0 \).
• Observing the graph's behavior as \( t \) approaches 0.

These observations can reveal the limit's value graphically even when algebraic methods become complex. Adjust the window carefully for accurate results, and use zooming features to scrutinize behavior closely near the limit point.