Question

Evaluate the following limits. $$\lim _{\theta \rightarrow 0} \frac{\frac{1}{2+\sin \theta}-\frac{1}{2}}{\sin \theta}$$

Short answer

Based on the given solution, create a short answer question: Question: Evaluate the limit of the following expression as 𝜃 approaches 0: $$\lim_{\theta\rightarrow 0}\frac{\frac{1}{2+\sin \theta}-\frac{1}{2}}{\sin \theta}$$ Answer: $$\frac{-1}{4}$$

Step-by-step solution

Rewrite the expression with a common denominator

To rewrite the expression with a common denominator, we need to consider the two fractions in the numerator: \(\frac{1}{2+\sin \theta}\) and \(\frac{1}{2}\). The common denominator for these two fractions will be \((2+\sin \theta)\times(2)\), and we will rewrite them accordingly: $$\frac{\frac{1}{2+\sin \theta}-\frac{1}{2}}{\sin \theta} = \frac{\frac{2 - (2+\sin\theta)}{2(2+\sin\theta)}}{\sin\theta} $$

Simplify the numerator

Now, we will simplify the numerator: $$\frac{2 - (2+\sin\theta)}{2(2+\sin\theta)} = \frac{-\sin\theta}{2(2+\sin\theta)}$$ So, the expression becomes: $$\frac{\frac{-\sin\theta}{2(2+\sin\theta)}}{\sin\theta} $$

Divide the expression by sin 𝜃

Next, we will divide the expression by sin 𝜃: $$\frac{-\sin\theta}{\sin\theta(2(2+\sin\theta))}$$

Simplify the expression further by cancelling terms

Now, we can cancel the common term \(\sin\theta\) from the numerator and denominator: $$\frac{-1}{2(2+\sin\theta)}$$

Evaluate the limit

Finally, we will evaluate the limit as 𝜃 approaches 0: $$\lim_{\theta\rightarrow 0}\frac{-1}{2(2+\sin\theta)} = \frac{-1}{2(2+\sin(0))} = \frac{-1}{2(2+0)} = \frac{-1}{4}$$ So, the limit $$\lim_{\theta\rightarrow 0}\frac{\frac{1}{2+\sin \theta}-\frac{1}{2}}{\sin \theta}$$ is equal to $$\frac{-1}{4}$$.

Key concepts

Trigonometric Limits

Trigonometric limits are essential in calculus, especially when dealing with functions involving trigonometric expressions. For this exercise, we need to evaluate a limit where trigonometric functions are part of the expression, specifically with \(\sin \theta\). These limits often involve small angle approximations or identities, but here, we use division and simplification instead.

Understanding trigonometric limits often includes recognizing common limits such as:

• \(\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1\)
• \(\lim_{\theta \to 0} \frac{1 - \cos \theta}{\theta} = 0\)

In this example, we perform algebraic manipulation to simplify the fraction and use these identities indirectly to solve the problem. Knowing these limits helps when approaching similar problems with trigonometric expressions.

Limit Evaluation

Limit evaluation is a process in calculus used to find the value that a function approaches as the input approaches a certain number. In this problem, we want to determine the behavior as \(\theta\) approaches zero. Evaluating limits often involves simplifying the expression or using strategies like L'Hôpital's Rule when direct substitution isn't possible.

Key steps in this evaluation include:

• Finding a common denominator to combine fractions and simplify the expression
• Dividing through by terms to isolate behaviors like \(\sin \theta\)
• Cancelling terms wisely to make the expression manageable

The final step typically involves substituting the limit value into the simplified expression. If simplification is done correctly, this will help in directly finding the limiting behavior.

Calculus Problem Solving

Calculus problem solving involves a series of logical steps that lead to understanding and solving complex problems. Each problem can often be broken down into manageable parts, just like solving the limit in this exercise.

To tackle similar problems effectively, you can:

• Rewrite complex expressions to look simpler using algebraic identities
• Break down the problem into smaller parts to simplify the overall expression
• Utilize known limit properties and trigonometric identities
• Constantly check intermediate steps to ensure accuracy in calculations

This process not only aids in solving limits but also builds a solid foundation for other areas in calculus. By following a structured approach, students can confidently solve problems and grasp the underlying concepts.