Problem 51
Question
\(\lim _{x \rightarrow 2}\left(\frac{\sqrt{1-\cos \\{2(x-2)\\}}}{x-2}\right)\) (A) equals \(\frac{1}{\sqrt{2}}\) (B) Does not exist (C) equals \(\frac{1}{\sqrt{2}}\) (D) equals \(-\sqrt{2}\)
Step-by-Step Solution
Verified Answer
The limit is \( \sqrt{2} \), but it does not match a choice, indicating a potential error in the options.
1Step 1: Examine the Limit
Consider the limit \( \lim_{x \to 2} \frac{\sqrt{1 - \cos{2(x-2)}}}{x-2} \). As \( x \to 2 \), the expression becomes indeterminate \( \frac{0}{0} \). Consequently, we need to apply an appropriate method to evaluate it.
2Step 2: Use Trigonometric Identity
Recall the trigonometric identity for small angles: \( 1 - \cos \theta \approx \frac{\theta^2}{2} \) for \( \theta \approx 0 \). Here, set \( \theta = 2(x-2) \), so \( 1 - \cos{2(x-2)} \approx \frac{(2(x-2))^2}{2} \).
3Step 3: Simplify using the Approximation
Substituting the approximation into the limit expression gives \( \lim_{x \to 2} \frac{\sqrt{\frac{(2(x-2))^2}{2}}}{x-2} \). This simplifies to \( \lim_{x \to 2} \frac{|2(x-2)|}{\sqrt{2}(x-2)} \).
4Step 4: Cancel Terms and Evaluate
Cancel \( (x-2) \) from the numerator and denominator, resulting in \( \lim_{x \to 2} \frac{2}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \).
5Step 5: Check the Correct Answer Choice
Comparing \( \sqrt{2} \) with the provided answer choices, none directly match \( \sqrt{2} \). Therefore, the limit simplifies to \( \sqrt{2} \), and the choice does not exist among the options.
Key Concepts
Indeterminate FormsTrigonometric LimitsTrigonometric Identities
Indeterminate Forms
Indeterminate forms occur in calculus when direct substitution in a limit expression results in an undefined form, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These forms require further analysis and manipulation to determine the true value of the limit.
In the given problem, as \( x \to 2 \), the expression \( \frac{\sqrt{1 - \cos{2(x-2)}}}{x-2} \) evaluates to \( \frac{0}{0} \). This signals an indeterminate form, implying the need for special techniques, often involving algebraic simplification or trigonometric identities, to resolve it.
Such forms are crucial in calculus as they help us explore the behavior of functions around critical points where the functions themselves don't smoothly yield answers. Various methods to resolve indeterminate forms include L'Hôpital's Rule, factoring, or using limits common to trigonometric functions.
In the given problem, as \( x \to 2 \), the expression \( \frac{\sqrt{1 - \cos{2(x-2)}}}{x-2} \) evaluates to \( \frac{0}{0} \). This signals an indeterminate form, implying the need for special techniques, often involving algebraic simplification or trigonometric identities, to resolve it.
Such forms are crucial in calculus as they help us explore the behavior of functions around critical points where the functions themselves don't smoothly yield answers. Various methods to resolve indeterminate forms include L'Hôpital's Rule, factoring, or using limits common to trigonometric functions.
Trigonometric Limits
Trigonometric limits involve evaluating the behavior of trigonometric functions as the variable approaches a certain value. These types of limits often require the use of trigonometric identities or approximations to simplify the problem.
In this exercise, the limit \( \lim_{x \to 2} \frac{\sqrt{1 - \cos{2(x-2)}}}{x-2} \) uses the approximations for small angles: \( 1 - \cos \theta \approx \frac{\theta^2}{2} \) when \( \theta \) is near zero. Understanding this identity allows us to replace expressions with more manageable forms that avoid complex original forms.
When approaching trigonometric limits, think about:
In this exercise, the limit \( \lim_{x \to 2} \frac{\sqrt{1 - \cos{2(x-2)}}}{x-2} \) uses the approximations for small angles: \( 1 - \cos \theta \approx \frac{\theta^2}{2} \) when \( \theta \) is near zero. Understanding this identity allows us to replace expressions with more manageable forms that avoid complex original forms.
When approaching trigonometric limits, think about:
- Identities that might simplify the expression.
- Common angles where sine or cosine has recognizable values.
- Approximations for small angles, useful for expressions involving \( \theta \to 0 \).
Trigonometric Identities
Trigonometric identities are the cornerstone of simplifying trigonometric expressions and solving limits involving trigonometric functions. They express relationships between the trigonometric functions and are essential tools for calculus problems.
In this context, we utilize the identity \( 1 - \cos \theta \approx \frac{\theta^2}{2} \) for small \( \theta \), which helps convert complicated expressions into simpler, equivalent forms. By setting \( \theta = 2(x-2) \), we are able to use this identity to replace \( 1 - \cos{2(x-2)} \) effectively.
Remember these tips while using trigonometric identities:
In this context, we utilize the identity \( 1 - \cos \theta \approx \frac{\theta^2}{2} \) for small \( \theta \), which helps convert complicated expressions into simpler, equivalent forms. By setting \( \theta = 2(x-2) \), we are able to use this identity to replace \( 1 - \cos{2(x-2)} \) effectively.
Remember these tips while using trigonometric identities:
- Always check if an identity can directly simplify the given problem.
- Use identities to rewrite the expression in a form that eliminates undesirable terms.
- Ensure the expression post-identity application is manageable with known calculus techniques.
Other exercises in this chapter
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