Question

Evaluate the limits with either L'Hôpital's rule or previously learned methods. $$ \lim _{x \rightarrow \pi / 2} \frac{\cos x}{2 \frac{\pi}{2}-x} $$

Short answer

The limit is 1.

Step-by-step solution

Recognize the Indeterminate Form

First, identify the form of the limit. Substitute \( x = \frac{\pi}{2} \) into the expression \( \frac{\cos x}{2 \frac{\pi}{2} - x} \). This yields \( \frac{0}{0} \), an indeterminate form, suggesting that L'Hôpital's rule can be applied.

Apply L'Hôpital's Rule

Since the limit is in \( \frac{0}{0} \) form, we can apply L'Hôpital's Rule, which states that \[ \lim _{x \rightarrow c} \frac{f(x)}{g(x)} = \lim _{x \rightarrow c} \frac{f'(x)}{g'(x)} \] provided the limit of the derivatives exists. Differentiate the numerator: \( \cos x \) becomes \( -\sin x \). Differentiate the denominator: \( 2 \frac{\pi}{2} - x \) becomes \( -1 \).

Evaluate the New Limit

Now, evaluate the limit of the new function \( \frac{-\sin x}{-1} = \sin x \) as \( x \to \frac{\pi}{2} \). \[ \lim_{x \to \frac{\pi}{2}} \sin x = 1 \].

State the Final Result

The limit evaluates to 1 based on the application of L'Hôpital's Rule and the fact that \( \sin \frac{\pi}{2} = 1 \).

Key concepts

L'Hôpital's Rule

L'Hôpital's Rule is a valuable tool in calculus for solving certain indeterminate limit forms. When you encounter an expression like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), traditional methods of finding limits often fall short. This is where L'Hôpital's Rule comes in handy. It states that for the limit \( \lim_{x \to c} \frac{f(x)}{g(x)} \), if substituting \( x = c \) gives an indeterminate form, you can instead consider \( \lim_{x \to c} \frac{f'(x)}{g'(x)} \).

You differentiate the numerator and the denominator separately, then take the limit of this new ratio. It's important to ensure that the new form is not undefined, and the derivatives exist before application. Additionally, try alternative methods first, as applying this rule can be unnecessary if simpler techniques work.

Indeterminate Forms

Indeterminate forms are expressions that are not immediately clear in value, particularly when calculating limits. Commonly encountered forms include \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0 \cdot \infty \), \( \infty - \infty \), \( 0^0 \), \( 1^\infty \), and \( \infty^0 \).

In the original exercise, substituting \( x = \frac{\pi}{2} \) results in the indeterminate form \( \frac{0}{0} \). When this happens, it’s a signal that straightforward substitution does not work, and other methods, like algebraic manipulation or L'Hôpital's Rule, need to be used to simplify the expression or find the limit.

The goal in handling indeterminate forms during limit evaluation is to transform the expression into a determinate form, often revealing the limit value.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate at which a function is changing at any given point. When applying L'Hôpital's Rule, differentiation is crucial because you need to find the derivatives of both the numerator and the denominator.

For instance, from the exercise, differentiating the numerator, \( \cos x \), gives \( -\sin x \). The derivative of the denominator, \( 2 \frac{\pi}{2} - x \), simplifies to \( -1 \). These derivatives are what allow the application of L'Hôpital's Rule to move from an indeterminate form to something more workable.

Understanding differentiation and its rules is crucial not only for solving limits but for many other applications in calculus and mathematical analysis.

Trigonometric Limits

Trigonometric limits can often be tricky because trigonometric functions, like sine and cosine, have their unique properties. One of the known limits in calculus is \( \lim_{x \to \frac{\pi}{2}} \sin x = 1 \). Knowing these standard limits can greatly simplify the evaluation of more complex expressions.

In the provided solution, after applying L'Hôpital's Rule, the new expression was \( \sin x \), for which the limit as \( x \to \frac{\pi}{2} \) is already known to be 1. Utilizing such established limits allows you to quickly find the values without additional complicated calculations.

When working with trigonometric limits, being familiar with key limits and identities can be immensely helpful. It reduces work and highlights connections between different mathematical concepts.