Problem 32 - Calculus for Biology and Medicine

Question

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow-\pi / 2} \frac{1+\tan ^{2} x}{\sec ^{2} x} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Understand the Limit Expression

We need to evaluate the limit \( \lim _{x \rightarrow-\pi / 2} \frac{1+\tan ^{2} x}{\sec ^{2} x} \). The expression contains trigonometric functions: tangent and secant. Recall that \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) and \( \sec(x) = \frac{1}{\cos(x)} \).

Step 2: Simplify the Expression Using Identities

We use a trigonometric identity where \(1 + \tan^2(x) = \sec^2(x)\). Substituting this identity into the expression, it becomes: \( \frac{\sec^2(x)}{\sec^2(x)} = 1\).

Step 3: Evaluate the Limit as x Approaches \(-\pi/2\)

Now that we have simplified the expression to a constant \(1\), the limit of \(1\) as \(x\) approaches \(-\pi/2\) is simply \(1\). As the expression is already constant, the value of the limit is unaffected by the value that \(x\) approaches.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental in calculus and help simplify complex expressions. One crucial identity used in calculus is the Pythagorean identity, which states that for any angle \(x\), \(1 + \tan^2(x) = \sec^2(x)\).

• Pythagorean identities are derived from the Pythagorean theorem, focusing on relationships between sine, cosine, and tangent functions. These identities help establish connections between different trigonometric functions.
• Simplification using these identities can transform elaborate expressions into simpler forms, making it easier to evaluate limits or derivatives. In our problem, we used the identity \(1 + \tan^2(x) = \sec^2(x)\) to simplify the given expression.

Remember: Simplifying expressions with identities is a powerful tool. It not only makes the math easier but also highlights the inherent relationships among trigonometric functions.

Tangent Function

The tangent function, denoted as \(\tan(x)\), is an essential trigonometric function that reveals a unique perspective on angles and ratios. It's defined as the ratio of sine to cosine:

• \(\tan(x) = \frac{\sin(x)}{\cos(x)}\)

Key Characteristics:

• The tangent function is periodic with a period of \( \pi\). This means it repeats its values every \( \pi\) radians.
• It has vertical asymptotes where \( \cos(x) = 0\), as the denominator of the function becomes zero, leading to undefined values. For example, \(x = \frac{\pi}{2}\), \(x = -\frac{\pi}{2}\), are points of undefined behavior.

Understanding these properties aids in comprehending the behavior of trigonometric functions near critical points and applying them effectively in calculus problems.

Secant Function

The secant function, represented as \(\sec(x)\), is another valuable trigonometric function, closely linked to the cosine function. It is defined as the reciprocal of the cosine:

• \(\sec(x) = \frac{1}{\cos(x)}\)

Notable Features:

• Like the tangent function, the secant has asymptotes, specifically where the cosine function equals zero. These occur at points like \(x = \frac{\pi}{2}\), \(x = -\frac{\pi}{2}\), etc.
• The secant function is also periodic and shares a period with the tangent function, repeating every \( \pi\) radians.

In calculus, recognizing these features of the secant function helps solve limits and understand the behavior of functions as they approach points of discontinuity.