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Magazine Chapter 3: Problem 50
Find the limits in Exercises \(47-54\) $$ \lim _{\theta \rightarrow \pi / 4} \frac{\tan \theta-1}{\theta-\frac{\pi}{4}} $$
Short Answer
Expert verified
The limit is 2.
Step by step solution
01
Identify Form of the Limit
The given expression is a limit problem of the form \( \lim _{\theta \rightarrow a} \frac{f(\theta)}{g(\theta)} \). Here, \( f(\theta) = \tan \theta - 1 \) and \( g(\theta) = \theta - \frac{\pi}{4} \) as \( \theta \rightarrow \frac{\pi}{4} \). We need to determine if this limit results in an indeterminate form like \( \frac{0}{0} \).
02
Check for Indeterminate Form
Substitute \( \theta = \frac{\pi}{4} \) into \( f(\theta) \) and \( g(\theta) \):- \( f(\frac{\pi}{4}) = \tan \frac{\pi}{4} - 1 = 1 - 1 = 0 \)- \( g(\frac{\pi}{4}) = \frac{\pi}{4} - \frac{\pi}{4} = 0 \)Both \( f(\theta) \) and \( g(\theta) \) are zero at \( \theta = \frac{\pi}{4} \), giving an indeterminate form \( \frac{0}{0} \). Therefore, L'Hospital's Rule can be applied.
03
Apply L'Hospital's Rule
L'Hospital's Rule states that if a limit is in the form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), the limit of \( \frac{f(\theta)}{g(\theta)} \) as \( \theta \to a \) can be found by evaluating \( \lim _{\theta \rightarrow a} \frac{f'(\theta)}{g'(\theta)} \).
04
Derivatives of the Functions
Find the derivatives of \( f(\theta) \) and \( g(\theta) \):- \( f'(\theta) = (\tan \theta - 1)' = \sec^2 \theta \)- \( g'(\theta) = (\theta - \frac{\pi}{4})' = 1 \)
05
Calculate the Limit Using Derivatives
Substitute the derivatives back into the limit:\[ \lim _{\theta \rightarrow \pi/4} \frac{f'(\theta)}{g'(\theta)} = \lim _{\theta \rightarrow \pi/4} \frac{\sec^2 \theta}{1} = \lim _{\theta \rightarrow \pi/4} \sec^2 \theta\]At \( \theta = \frac{\pi}{4} \), \( \sec \frac{\pi}{4} = \sqrt{2} \). Therefore, \( \sec^2 \frac{\pi}{4} = 2 \).
06
Conclusion
The limit simplifies to 2, so:\[ \lim _{\theta \rightarrow \pi/4} \frac{\tan \theta - 1}{\theta - \frac{\pi}{4}} = 2 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Indeterminate Forms
In calculus, indeterminate forms are expressions that do not have a clear or definite result when evaluating limits. The most common indeterminate forms are \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0 \times \infty \), \( \infty - \infty \), and others that don鈥檛 provide a straightforward conclusion.
Understanding these forms is crucial because they often indicate that the limit cannot be determined through direct substitution. Instead, special techniques like algebraic manipulation or calculus-based methods, such as L'Hospital鈥檚 Rule, are required.
Understanding these forms is crucial because they often indicate that the limit cannot be determined through direct substitution. Instead, special techniques like algebraic manipulation or calculus-based methods, such as L'Hospital鈥檚 Rule, are required.
- In our example, the limit expression \( \lim _{\theta \rightarrow \pi / 4} \frac{\tan \theta-1}{\theta-\frac{\pi}{4}} \) results in the \( \frac{0}{0} \) indeterminate form when directly substituting \( \theta = \frac{\pi}{4} \).
- This discovery allows us to employ techniques such as L鈥橦ospital鈥檚 Rule to evaluate the limit properly.
Limit Evaluation
Evaluating limits involves finding the value that a function approaches as the input approaches a given value. Many times, especially in calculus, limits are used to understand the behavior of functions that are not easily expressed at specific points.
Traditional limit evaluation through substitution might fail when encountering indeterminate forms. This is where calculus techniques like L'Hospital鈥檚 Rule become invaluable.
L'Hospital鈥檚 Rule can transform such tricky problems into more manageable ones by using derivatives. For example:
Traditional limit evaluation through substitution might fail when encountering indeterminate forms. This is where calculus techniques like L'Hospital鈥檚 Rule become invaluable.
L'Hospital鈥檚 Rule can transform such tricky problems into more manageable ones by using derivatives. For example:
- In the expression \( \lim _{\theta \rightarrow \pi / 4} \frac{\tan \theta-1}{\theta-\frac{\pi}{4}} \), direct substitution gives 0 in both the numerator and the denominator.
- Recognizing this indeterminate form allows us to apply L'Hospital鈥檚 Rule.
- By differentiating both the numerator and the denominator, the limit becomes \( \lim _{\theta \rightarrow \pi/4} \sec^2 \theta = 2 \).
- Thus, after evaluating the derivatives, the limit simplifies to 2.
Derivatives
Derivatives are fundamental in calculus, representing the rate of change of a function with respect to one of its variables. They form the backbone of methods such as L'Hospital's Rule, which is used to solve limit problems involving indeterminate forms.
Understanding how to take derivatives is crucial when applying L'Hospital鈥檚 Rule. This technique requires differentiating both the numerator and the denominator of a fraction to evaluate a limit.
Understanding how to take derivatives is crucial when applying L'Hospital鈥檚 Rule. This technique requires differentiating both the numerator and the denominator of a fraction to evaluate a limit.
- For our problem, the given functions are \( f(\theta) = \tan \theta - 1 \) and \( g(\theta) = \theta - \frac{\pi}{4} \).
- The derivatives are \( f'(\theta) = \sec^2 \theta \) and \( g'(\theta) = 1 \).
- These derivatives allow us to transform the original indeterminate form \( \frac{0}{0} \) into a simple evaluation of \( \sec^2 \theta \) as \( \theta \to \frac{\pi}{4} \).
- Thus, knowing derivatives enables the effective use of L'Hospital's Rule.
