Problem 76 - Calculus Early Transcendentals

Question

Evaluate the following limits or state that they do not exist. $$\lim _{x \rightarrow 0^{+}} \frac{1-\cos ^{2} x}{\sin x}$$

Step-by-Step Solution

Answer

Answer: The value of the limit is 0.

Step 1: Recognize the identity

Notice that the expression has \(1 - \cos^2x\), which is equal to \(\sin^2x\) according to the Pythagorean identity: \(\sin^2x + \cos^2x = 1\). So, rewrite the expression using this identity: $$\lim _{x \rightarrow 0^{+}} \frac{1-\cos ^{2} x}{\sin x} = \lim_{x \rightarrow 0^{+}}\frac{\sin^2x}{\sin x}$$

Step 2: Simplify the expression further

Now, we can cancel out a \(\sin x\) term as follows: $$\lim_{x \rightarrow 0^{+}}\frac{\sin^2x}{\sin x} = \lim_{x \rightarrow 0^{+}}\sin x$$

Step 3: Evaluate the limit

The limit we have now is a simple limit involving the sine function: $$\lim_{x \rightarrow 0^{+}}\sin x$$ As x approaches 0 from the positive side, the sine function approaches 0. Thus, the limit evaluates to: $$\lim_{x \rightarrow 0^{+}}\sin x = 0$$ So, the final answer is: $$\lim _{x \rightarrow 0^{+}} \frac{1-\cos ^{2} x}{\sin x} = 0$$

Key Concepts

Pythagorean Identity

Understanding the Pythagorean identity is crucial when dealing with trigonometric expressions in calculus. This fundamental principle reveals a relationship between the sine and cosine of an angle, stating that \(\sin^2{\theta} + \cos^2{\theta} = 1\). When evaluating trigonometric limits, you can often use the Pythagorean identity to simplify expressions.
In the given problem \( 1 - \cos^2{x} \) on the numerator can be transformed into \( \sin^2{x} \) because \( \sin^2{x} + \cos^2{x} = 1 \) by the identity. This simplification often opens the path to further simplification or direct evaluation of the limit.

Simplifying Expressions

Simplifying expressions is a key step in solving calculus problems, especially when evaluating limits. The essence of simplification is to make complex expressions more manageable, which often involves canceling common terms, factoring, combining like terms, or using algebraic identities.
In our example, after applying the Pythagorean identity, we simplify by canceling out \( \sin x \) from both the numerator and the denominator, reducing \( \frac{\sin^2{x}}{\sin x} \) to \( \sin x \). Such simplifications can make limits much more straightforward to evaluate and can turn an intractable expression into one that is easy to manage.

Trigonometric Limits

Trigonometric limits are an important category in calculus as they involve angles and their respective trigonometric functions. Evaluating limits involving trigonometric functions requires knowledge of certain limits (like \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \) ) and identities.
When approaching 0, the sine function, as in \( \lim_{x \rightarrow 0^{+}}\sin x \), the behavior of the function is predictable—it approaches 0. Understanding how these functions behave as they approach certain values is imperative in correctly evaluating trigonometric limits.

Limit Evaluation Methods

There are various methods to evaluate limits in calculus, such as direct substitution, factoring, rationalization, and using L'Hôpital's rule for indeterminate forms. Another important method is simplification, prominently used when direct substitution isn't possible.
In the sample problem, after simplification, direct substitution is used because \( \sin x \) is continuous at \( x = 0 \), and we can simply substitute \( x \) with 0 to find that \( \lim_{x \rightarrow 0^{+}}\sin x = 0 \). By mastering different evaluation methods, you can approach a variety of limits with confidence and find the right tool for the job.