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: Identify the Trigonometric Identity

We start by recognizing the trigonometric identity that relates tangent and secant. Recall that the identity is \(1 + \tan^2 x = \sec^2 x\). We can use this identity to simplify the given limit.

Step 2: Simplify the Expression with the Identity

Using the identity from Step 1, substitute \( \sec^2 x \) for \( 1 + \tan^2 x \) in the limit expression. This gives us: \[\lim_{x \to -\pi/2} \frac{\sec^2 x}{\sec^2 x}\] This simplifies to: \[\lim_{x \to -\pi/2} 1\]

Step 3: Evaluate the Simplified Limit

The expression \( \lim_{x \to -\pi/2} 1 \) is straightforward because it represents a constant function. The limit of a constant is simply the constant itself.

Key Concepts

Trigonometric Identities

In calculus, trigonometric identities play a crucial role, especially when working with limits. These identities are equations involving trigonometric functions that are true for every value of the involved variables. A key identity used in many limit problems is the Pythagorean identity:

• \(1 + \tan^2 x = \sec^2 x\)

This identity is derived from the common Pythagorean identity, \(\sin^2 x + \cos^2 x = 1\). Knowing these identities can significantly simplify the expression you’re dealing with.
For example, in our initial problem, understanding that the expression \(1 + \tan^2 x\) is equal to \(\sec^2 x\) allows us to easily substitute and simplify the expression. Hence, good knowledge of these identities is a powerful tool for anyone learning calculus.

Limit Evaluation

Limit evaluation is a fundamental concept in calculus, and it involves finding the value that a function approaches as the input (or variable) approaches some value. When evaluating limits, especially those involving trigonometric functions, substituting known identities often simplifies the task.
In the given exercise, we are evaluating the limit as \(x\) approaches \(-\pi/2\). Through simplification, it becomes a constant function.

• Direct substitution is the process of plugging the value into the simplified function.
• Since our expression \(\frac{\sec^2 x}{\sec^2 x}\) simplifies to \(1\), the limit is evaluated as \(1\).

By recognizing this constant, we make limit evaluation a straightforward process. Evaluating limits in such linearly simplified contexts confirms the direct relationship between simplification and easy evaluation.

Simplifying Expressions

Simplifying expressions is all about making a complex expression easier to work with. This can involve combining like terms, factoring, or using identities. A simplified expression not only looks tidier, but it’s also often much easier to work with in further calculus operations.
The importance of simplifying cannot be overstated. In the original problem, simplifying \(\frac{1+\tan^2 x}{\sec^2 x}\) to \(\frac{\sec^2 x}{\sec^2 x}\) shows an immediate simplification to \(1\).

• Such simplification instantly clarifies that no further calculations are needed to find the limit.
• This reduction from a seemingly complex fraction to a constant illustrates the power of recognizing and applying identities.

Always remember, the simpler the expression, the easier it is to figure out its behavior, such as finding its limit.