Problem 26 - University Calculus: Early Transcendentals

Question

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow(\pi / 2)}\left(\frac{\pi}{2}-x\right) \tan x$$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify the Indeterminate Form

First, we need to evaluate the limit expression to see if it results in an indeterminate form. As \( x \) approaches \( \frac{\pi}{2} \), \( \tan x \) approaches infinity since tangent is undefined at \( \frac{\pi}{2} \). Thus, \( \left(\frac{\pi}{2} - x\right) \tan x \) results in an indeterminate form of \( \frac{0 \cdot \infty}{1} \) or potentially \( \infty \cdot 0 \). This form allows us to use L'Hôpital's Rule.

Step 2: Transform the Expression

Reformulate the expression to fit a type where L'Hôpital's Rule applies, typically \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). We rewrite the expression \( \left(\frac{\pi}{2} - x\right) \tan x \) as the fraction \( \frac{\pi/2 - x}{\cot x} \). This fraction approaches the form \( \frac{0}{0} \) as \( x \to \frac{\pi}{2} \).

Step 3: Apply L'Hôpital's Rule

L'Hôpital's Rule states that for a limit of the form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), we can differentiate the numerator and the denominator separately. Differentiating the numerator, \( \frac{d}{dx} \left( \frac{\pi}{2} - x \right) = -1 \). Differentiating the denominator, \( \frac{d}{dx}(\cot x) \) gives \( -\csc^2 x \). Thus, the limit \( \lim _{x \rightarrow(\pi / 2)} \frac{\frac{\pi}{2} - x}{\cot x} \) becomes \( \lim_{x \rightarrow(\pi / 2)} \frac{-1}{-\csc^2 x} = \lim_{x \rightarrow(\pi / 2)} \csc^2 x \).

Step 4: Evaluate the Limit After Differentiation

Now we evaluate the limit \( \lim_{x \rightarrow(\pi / 2)} \csc^2 x \). Recall that \( \csc x = \frac{1}{\sin x} \). As \( x \) approaches \( \frac{\pi}{2} \), \( \sin x \) approaches 1, hence \( \csc x \) also approaches 1, and \( \csc^2 x \) approaches 1. Therefore, the evaluated limit is 1.

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a handy tool in limit calculations, particularly when dealing with indeterminate forms. The rule asserts that for expressions resulting in indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), one can differentiate the numerator and the denominator separately to resolve the limit.

However, it's vital to ensure that the new expression, after differentiation, is not in an indeterminate form anymore or continues to resolve correctly.

• Works well for limits in the form of \( \frac{f(x)}{g(x)} \)
• Differentiate \( f(x) \) and \( g(x) \) separately
• Re-evaluate the limit

Utilizing L'Hôpital's Rule can simplify otherwise complex expressions and make limit evaluation straightforward.

Indeterminate Forms

Indeterminate forms arise when trying to evaluate limits that do not initially resolve to a real number right away. Common indeterminate forms include \( \frac{0}{0}, \infty - \infty, 0 \cdot \infty, \frac{\infty}{\infty}, 0^0, \infty^0\), and so on.

These forms do not provide a definite result, which is why further analytical techniques, such as L'Hôpital's Rule, are used to simplify and find the limits.

• Occur when direct substitution leads to undefined results
• Requires rewriting or transformation for evaluation

Understanding indeterminate forms is critical to choosing the right approach for solving limit problems.

Differentiation

Differentiation is a fundamental concept in calculus, involving the computation of derivatives, which represent rates of change or slopes of functions. For instance, finding the derivative of \( f(x) \) involves determining \( f'(x) \), the function expressing its instantaneous rate of change for any given \( x \).

This technique is crucial when applying L'Hôpital's Rule, since both the numerator and denominator of an expression are differentiated separately.

• Find the derivative to assess the rate of change
• Gives us a new function to work with in limit calculations

This differentiation transforms complex limits into simpler forms that are easier to evaluate.

Trigonometric Limits

Understanding trigonometric limits is essential when working with limits involving trigonometric functions like sine, cosine, and tangent. These limits often require specific techniques due to their periodic nature and unique properties.

For example, knowing that \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \) and \( \lim_{x \to 0} \frac{1-\cos x}{x} = 0 \) is crucial in solving many limit problems involving trigonometric functions.

• Focus on key limits involving sine, cosine, and tangent
• Use identities to simplify problems

In the context of L'Hôpital's Rule, understanding these limits allows for the reduction of seemingly complex terms to simpler calculable forms.