Problem 48
Question
Find the limits. $$\lim _{x \rightarrow-\pi / 6} \sqrt{1+\cos (\pi \csc x)}$$
Step-by-Step Solution
Verified Answer
The limit is \( \sqrt{2} \).
1Step 1: Understand the Function
The function given is \( \sqrt{1+\cos(\pi \csc x)} \). Our task is to find the limit as \( x \) approaches \(-\frac{\pi}{6}\).
2Step 2: Substitute \( x = -\frac{\pi}{6} \) into \( \csc x \)
Since \( \csc x = \frac{1}{\sin x} \), substitute \( x = -\frac{\pi}{6} \) to get \( \sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2} \). Therefore, \( \csc\left( -\frac{\pi}{6} \right) = \frac{-1}{-1/2} = -2 \).
3Step 3: Substitute \( \csc x = -2 \) into \( \cos(\pi \csc x) \)
Now substitute \( \csc x = -2 \) into \( \cos(\pi \csc x) \) to get \( \cos(-2\pi) \). Since \( \cos(-\theta) = \cos(\theta) \), this becomes \( \cos(2\pi) \).
4Step 4: Evaluate \( \cos(2\pi) \)
\( \cos(2\pi) = 1 \) because cosine of any integer multiple of \( 2\pi \) is 1.
5Step 5: Calculate \( \sqrt{1 + \cos(2\pi)} \)
Substitute the value from Step 4 into the original function: \( \sqrt{1 + 1} = \sqrt{2} \).
6Step 6: Conclude the Limit Evaluation
Thus, the limit of the function as \( x \) approaches \(-\frac{\pi}{6}\) is \( \sqrt{2} \).
Key Concepts
Trigonometric LimitsTrigonometric IdentitiesLimit Evaluation Techniques
Trigonometric Limits
Understanding trigonometric limits can be both interesting and rewarding. They often involve understanding how trigonometric functions behave as the variable approaches a certain point. In calculus, limits help us find where a function or expression is heading as one or more of its variables approach certain values.
For the exercise given, we are examining the limit of the function \( \lim_{x \rightarrow -\pi / 6} \sqrt{1+\cos(\pi \csc x)} \).This means we want to discover the value that the function approaches when \( x \) gets very close to \(-\frac{\pi}{6}\).
When evaluating, we are particularly interested in how the inner trigonometric transformation, in this case, \( \pi \csc x \), changes as \( x \) gets near \(-\frac{\pi}{6}\). This change dictates the behavior of the resulting expression as a whole.Identifying these limits is crucial for understanding continuity, differentiability, and the behavior of trigonometric expressions in calculus.
For the exercise given, we are examining the limit of the function \( \lim_{x \rightarrow -\pi / 6} \sqrt{1+\cos(\pi \csc x)} \).This means we want to discover the value that the function approaches when \( x \) gets very close to \(-\frac{\pi}{6}\).
When evaluating, we are particularly interested in how the inner trigonometric transformation, in this case, \( \pi \csc x \), changes as \( x \) gets near \(-\frac{\pi}{6}\). This change dictates the behavior of the resulting expression as a whole.Identifying these limits is crucial for understanding continuity, differentiability, and the behavior of trigonometric expressions in calculus.
Trigonometric Identities
Trigonometric identities are essential tools in mathematics that simplify expressions and solve equations involving trigonometric functions.These identities allow us to express one trigonometric function in terms of others,which can be helpful when evaluating limits.
For instance, let's consider the cosecant function, \( \csc x \). The identity \( \csc x = \frac{1}{\sin x} \) allows us to understand and substitute reciprocal values involved in the problem. In the step-by-step solution, recognizing that \( \sin(-\frac{\pi}{6}) = -\frac{1}{2} \) directly led us to calculate \( \csc(-\frac{\pi}{6}) = -2 \).
Clever application of such identities leads to simpler expressions. This in turn suits effortless evaluation of limits. Remember too that the identity \( \cos(-\theta) = \cos(\theta) \) simplified our substitution \( \cos(-2\pi) \) to \( \cos(2\pi) \), confirming that its value remains consistent throughout the periodic cycles of the cosine function.
For instance, let's consider the cosecant function, \( \csc x \). The identity \( \csc x = \frac{1}{\sin x} \) allows us to understand and substitute reciprocal values involved in the problem. In the step-by-step solution, recognizing that \( \sin(-\frac{\pi}{6}) = -\frac{1}{2} \) directly led us to calculate \( \csc(-\frac{\pi}{6}) = -2 \).
Clever application of such identities leads to simpler expressions. This in turn suits effortless evaluation of limits. Remember too that the identity \( \cos(-\theta) = \cos(\theta) \) simplified our substitution \( \cos(-2\pi) \) to \( \cos(2\pi) \), confirming that its value remains consistent throughout the periodic cycles of the cosine function.
Limit Evaluation Techniques
Evaluating limits, particularly trigonometric ones, involves using specific techniques to simplify expressions.Here we discuss some key strategies used to solve the original exercise and others like it.
1. **Substitution Method:**Direct substitution is one of the simplest techniques used when the function is defined at the limit point.In our step-by-step approach, we replaced \( x \) with \(-\frac{\pi}{6} \) to evaluate \( \csc x \), leading to further simplifications.
2. **Trigonometric Identities:**Utilizing identities can transform or simplify trigonometric expressions, as showcased in the problem with \( \csc(x) \) and \( \cos(-\theta) \). By applying correct identities, the limit turned into a more manageable form.
3. **Handling Indeterminate Forms:**Some limits lead to indeterminate forms like \( \frac{0}{0} \). If encountered, further steps like l'Hôpital's Rule or series expansion might be necessary.
For our example, such complexities did not arise. Yet, understanding these fallback techniques ensures comprehensive capabilities for handling any limits entailing trigonometric functions.
1. **Substitution Method:**Direct substitution is one of the simplest techniques used when the function is defined at the limit point.In our step-by-step approach, we replaced \( x \) with \(-\frac{\pi}{6} \) to evaluate \( \csc x \), leading to further simplifications.
2. **Trigonometric Identities:**Utilizing identities can transform or simplify trigonometric expressions, as showcased in the problem with \( \csc(x) \) and \( \cos(-\theta) \). By applying correct identities, the limit turned into a more manageable form.
3. **Handling Indeterminate Forms:**Some limits lead to indeterminate forms like \( \frac{0}{0} \). If encountered, further steps like l'Hôpital's Rule or series expansion might be necessary.
For our example, such complexities did not arise. Yet, understanding these fallback techniques ensures comprehensive capabilities for handling any limits entailing trigonometric functions.
Other exercises in this chapter
Problem 48
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