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: Understanding the Expression

We are given the expression \( \lim_{x \rightarrow \pi / 2} \frac{\cos^2 x}{1 - \sin^2 x} \). Recognize that \(1 - \sin^2 x\) can be rewritten using the Pythagorean identity \(\cos^2 x = 1 - \sin^2 x\).

Step 2: Simplify the Denominator

Rewrite the denominator \(1 - \sin^2 x\) using the identity: \( \cos^2 x = 1 - \sin^2 x \). This simplifies the expression to \( \lim_{x \rightarrow \pi / 2} \frac{\cos^2 x}{\cos^2 x} \).

Step 3: Evaluate the Simplified Expression

Simplifying \( \frac{\cos^2 x}{\cos^2 x} \) gives \( 1 \) for \(x eq \frac{\pi}{2}\). However, since we are taking the limit as \( x \rightarrow \frac{\pi}{2} \), we only need to ensure the expression was correctly simplified for all x near \( \frac{\pi}{2} \).

Step 4: Consider the Limit Result

Since the expression \( \frac{\cos^2 x}{\cos^2 x} \) equals \( 1 \) everywhere it's defined around \( x = \frac{\pi}{2} \) (except potentially at \( x = \frac{\pi}{2} \) where \( \cos x = 0 \)), the limit evaluates to \( 1 \) as \( x\) approaches \( \frac{\pi}{2} \).

Key Concepts

Pythagorean identity

One of the most beautiful aspects of trigonometry is the relationship between the sine and cosine functions.This relationship is encapsulated in the Pythagorean identity, which states: \(\cos^2 x + \sin^2 x = 1\).This identity is incredibly useful when working through problems involving trigonometric functions.Let's see how this applies to our limit problem.Given our expression, \(1 - \sin^2 x\) in the denominator can be rewritten using the Pythagorean identity.Rewriting it as \(\cos^2 x\) helps us simplify the expression significantly.

• Recognize the relationship: \(\cos^2 x + \sin^2 x = 1\).
• Simplify expressions by replacing terms using this identity.
• Makes complex expressions more manageable.

This identity serves as a foundational tool in simplifying trigonometric limits.

limit evaluation

Evaluating limits involves determining the value that a function approaches as the variable approaches a particular point.In this exercise, the task is to find the limit of an expression as \(x\) approaches \(\frac{\pi}{2}\).Here, substituting \(1 - \sin^2 x\) with \(\cos^2 x\) allowed for drastic simplification of the limit expression.After simplifying, the expression reduces to \(\frac{\cos^2 x}{\cos^2 x}\).This is simply 1 for all \(x\) except where division by zero occurs.

To evaluate accurately:

• Simplify first using algebraic identities (like the Pythagorean identity).
• Check for points where the function might be undefined (e.g., division by zero).
• Analyze behavior around the limit point to ensure correctness.

The limit evaluation brings clarity, especially in cases where the direct substitution might cause undefined results.This method allows you to step around discontinuity and zero in on the behaviors of functions closely.

trigonometric simplification

Trigonometric simplification is the process of using trigonometric identities to convert complex expressions into simpler, more manageable forms.In our problem, using the Pythagorean identity was an instance of such a simplification.By rewriting \(1 - \sin^2 x\) as \(\cos^2 x\), the initial complex expression became much easier to evaluate.Simplification can often involve the following techniques:

• Rewriting terms using basic trigonometric identities like \(\cos^2 x = 1 - \sin^2 x\).
• Combining similar terms to reduce complexity.
• Checking for opportunities to cancel terms across numerator and denominator.

Reducing complexity in mathematical expressions is essential for clearer understanding and easier computations.It is an important skill not only in limits but throughout calculus and beyond.