Problem 31 - Calculus for Biology and Medicine

Question

In Problems \(29-48\), find the limits. $$ \lim _{x \rightarrow \pi / 2} \frac{\cos ^{2} x}{1-\sin ^{2} x} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify the Limit Expression

First, let's identify the expression where the limit is to be found: \( \lim _{x \rightarrow \pi / 2} \frac{\cos ^{2} x}{1-\sin ^{2} x} \). This expression includes trigonometric functions, so careful substitution around the limit point is necessary since \( \pi/2 \) involves common trigonometric values.

Step 2: Simplify the Denominator

Observe that the denominator can be simplified using the Pythagorean identity: \( 1 - \sin^2 x = \cos^2 x \). Therefore, the expression becomes \( \frac{\cos^2 x}{\cos^2 x} \).

Step 3: Simplify the Overall Expression

Now, we can simplify the expression \( \frac{\cos^2 x}{\cos^2 x} \) to 1, as the numerator and denominator are identical. Thus, the expression simplifies to just 1 for all \( x eq \pi/2 \) (since \( \cos(\pi/2) = 0 \)).

Step 4: Evaluate the Limit

Since the expression \( \frac{\cos^2 x}{1-\sin^2 x} \) simplifies to 1 everywhere around \( x = \pi/2 \) where it's defined, the limit as \( x \) approaches \( \pi/2 \) is simply 1.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is crucial when dealing with expressions that have sine and cosine functions. These identities are equations involving trigonometric functions that are true for all values of the variables involved. They help simplify expressions and solve trigonometric equations effectively.
To tackle the problem given, we used one of the basic trigonometric identities known as the Pythagorean identity:

• \( 1 - \sin^2 x = \cos^2 x \)

This identity stems from the Pythagorean theorem and is crucial in numerous limit and trigonometric problems. By using this identity, we were able to transform the denominator of the original expression from \(1 - \sin^2 x\) to \(\cos^2 x\). This simplification aligns the denominator with the numerator, thus enabling further simplification.
Getting comfortable with common trigonometric identities like the above ensures not just mastery in limits but also enables you to approach a wide range of calculus problems with ease.

Limit Evaluation

Limits are a fundamental concept in calculus which measure the value a function approaches as the input gets closer to a particular point. In this problem, we're interested in finding the limit as \( x \) approaches \( \pi/2 \) of the expression \( \frac{\cos^2 x}{1-\sin^2 x} \).
Evaluating limits often involves substituting the value directly into the function. However, this direct method requires that substitutions do not result in undefined expressions, like division by zero. In our case, direct substitution at \( x = \pi/2 \) evaluates the expression as \( \frac{0}{0} \), which is indeterminate. That's why further investigation and simplification are crucial to evaluate the limit properly.
Limits help find the behavior of functions near certain points, which leads to understanding continuity and differentiability later on. This specific evaluation of the limit where both the numerator and the denominator become zero is common and necessitates simplifying techniques.

Simplifying Expressions

Simplifying expressions can greatly assist with limit evaluations and other calculus problems. It involves reducing the expressions to their simplest form while retaining equivalent value or behavior. In the given problem, simplification was key.
Our original problem was \( \frac{\cos^2 x}{1-\sin^2 x} \). By recognizing and applying the relevant trigonometric identity, we simplified the denominator to \( \cos^2 x \), which matches the numerator. Thus, the expression simplifies down to \( 1 \), providing direct insight into the limit.
The ability to simplify effectively:

• Reduces computational complexity
• Allows for immediate profiling of limits
• Aids in seeing straightforward patterns in trigonometric, polynomial, and other types of functions

Learning to simplify expressions not only aids in solving the current problem but also builds confidence in tackling complex calculus questions universally.