Problem 23 - Calculus for Biology and Medicine

Question

Use l'Hospital's rule to find the limits. $$ \lim _{\rightarrow(\pi / 2)-} \frac{\tan x}{\sec ^{2} x} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Verify Indeterminate Form

Substitute \( x = \frac{\pi}{2} \) in the given function \( \frac{\tan x}{\sec^2 x} \). We get \( \tan\left(\frac{\pi}{2}\right) = \infty \) and \( \sec^2\left(\frac{\pi}{2}\right) = \infty \), resulting in an indeterminate form \( \frac{\infty}{\infty} \).

Step 2: Apply L'Hôpital's Rule

Since the limit results in the \( \frac{\infty}{\infty} \) form, we apply L'Hôpital's Rule. Differentiate the numerator and the denominator with respect to \( x \). For \( \tan x \), the derivative is \( \sec^2 x \), and for \( \sec^2 x \), the derivative is \( 2 \sec^2 x \tan x \).

Step 3: Write New Limit

The new limit to evaluate is: \[ \lim _{x \rightarrow (\pi / 2)^-} \frac{\sec^2 x}{2 \sec^2 x \tan x} \] This simplifies to: \[ \lim _{x \rightarrow (\pi / 2)^-} \frac{1}{2 \tan x} \]

Step 4: Evaluate the Simplified Limit

Substitute \( x = \frac{\pi}{2} \) in the simplified expression \( \frac{1}{2 \tan x} \). This becomes \( \frac{1}{2 \times \infty} = 0 \). As \( x \) approaches \( \frac{\pi}{2} \) from the left, \( \tan x \) approaches infinity, which results in the limit going to zero.

Key Concepts

Indeterminate Forms

When working with limits in calculus, you might encounter expressions that don't initially make much sense. An indeterminate form is one such expression where substituting directly into the equation doesn't give you a clear answer. Common types of indeterminate forms include

• \(\frac{0}{0}\)
• \(\frac{\infty}{\infty}\)
• \(\infty - \infty\)
• \(0 \cdot \infty\)

These forms are called 'indeterminate' because, without additional techniques, you can't predict the behavior of the function as it approaches a certain point. In the problem you're working on, substituting directly resulted in \(\frac{\infty}{\infty}\), which is a classic indeterminate form.
To handle this situation, we use special strategies such as L'Hôpital's Rule, which allows us to find limits by differentiating the numerator and denominator.

Calculus Limits

Calculus often deals with the concept of limits, which ask us what value a function approaches as the input gets infinitely close to a certain number. Limits are fundamental because they allow us to rigorously define many concepts and calculations in calculus, including derivatives and integrals.
In your exercise, the limit was evaluated as \(x\) tended towards \(\frac{\pi}{2}\) from the left, which is denoted as \(x \to (\frac{\pi}{2})^-\). This notation means that we're interested in the values of \(x\) that are slightly less than \(\frac{\pi}{2}\), focusing on how the function behaves as we approach from one direction.
Solving the limit required transforming the original function using derivatives because its direct substitution offered no clear answers. By differentiating, we were able to simplify and then evaluate the limit successfully.

Differentiation

Differentiation is a key concept in calculus, referring to the process of finding the derivative of a function. The derivative represents the rate of change or slope of the function at any given point. In the context of limits, differentiation can help simplify expressions that initially seem unsolvable due to indeterminate forms.
In the exercise, the derivatives of \(\tan x\) and \(\sec^2 x\) were calculated to apply L'Hôpital's Rule.

• The derivative of \(\tan x\) is \(\sec^2 x\).
• The derivative of \(\sec^2 x\) turns out to be \(2 \sec^2 x \tan x\).

By differentiating the numerator and the denominator, a new limit \(\frac{1}{2 \tan x}\) was obtained. Simplifying the problem this way made it possible to evaluate the limit as \(x\) moved towards \(\frac{\pi}{2}\).
Differentiation not only simplified the calculation but also transformed the complex expression into one that was easier to approach and solve.