Question

Use analytic methods to show that \(\lim _{h \rightarrow 0} \frac{\cos h-1}{h}=0\)

Short answer

The limit as \( h \rightarrow 0 \) of the function \( \frac{\cos h-1}{h} \) is 0.

Step-by-step solution

Identify the limit

First, identify what we are trying to find. We are looking at the limit as \(h\) approaches 0 for the function \( \frac{\cos h-1}{h}\).

Simplify the function

Next, simplify the function. We can use the limit definition of the derivative to recognize that \( \frac{f(x + h) - f(x)}{h} \) as \( h \rightarrow 0 \) is the derivative of \( f(x) \) with respect to \( x \). So, in our case, \( f(x) = \cos x \). Therefore, the limit of the function as \( h \rightarrow 0 \) is the derivative of \( \cos x \), when \( x = 0 \).

Compute the derivative

The derivative of \( \cos x \) is \( -\sin x \). So the derivative of \( \cos x \) at \( x = 0 \) is \( -\sin(0) \).

Evaluate the limit

As \( \sin(0) = 0 \), \(-\sin(0) = 0\). Therefore, the limit as \( h \rightarrow 0 \) of the function \( \frac{\cos h-1}{h} \) is 0.

Key concepts

Analytic Methods in Calculus

In calculus, analytic methods encompass a variety of techniques used to evaluate limits, derivatives, integrals, and series. Specifically, when examining the behavior of functions as variables approach certain values, limits are fundamental.

For the given exercise, \(\lim _{h \rightarrow 0} \frac{\cos h-1}{h}=0\), an analytical method involves recognizing that the limit can be connected to the concept of a derivative. Here, the function \( \frac{\cos h-1}{h} \) closely resembles the definition of the derivative, which is the essence of its analytic evaluation.

As a piece of exercise improvement advice, it's crucial to link the problem to the fundamental principles of calculus. In this case, identifying the expression as a derivative at an early stage simplifies the process. Understanding these connections not only helps in solving the immediate problem but also in recognizing patterns in more complex problems and enhancing analytical skills in calculus.

Derivative of Cosine

The derivative is a concept that measures how a function changes as its input changes. The cosine function, \(\cos x\), is a trigonometric function that is particularly interesting because its behavior is periodic, repeating every \(2\pi\) radians.

To find the derivative of the cosine function, you apply the limit definition of the derivative. To elaborate, for any function \(f(x)\), the derivative \(f'(x)\) can be found using the limit expression \(\lim _{h \rightarrow 0} \frac{f(x + h) - f(x)}{h}\). For \(f(x) = \cos x\), it leads to the derivative being \( -\sin x \).

It's important when learning this concept, to practice finding the derivatives of trigonometric functions, which often come up in calculus. Understanding the derivative of cosine is not just about memorizing that it equals \( -\sin x \), but also comprehending how it's derived from the limit definition, as shown in the exercise.

L'Hôpital's Rule

L'Hôpital's Rule is a widely used tool in calculus for finding limits that present an indeterminate form, such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). The rule states that if the limits of the functions \(f(x)\) and \(g(x)\) both approach 0 or both approach infinity as \(x\) approaches a certain value, and the derivative of \( f \) and \( g \) are continuous at this point, then \(\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}\).

In the context of our exercise, L'Hôpital's Rule could have been applied if we faced an indeterminate form when evaluating the limit of \(\frac{\cos h-1}{h}\) as \(h\) approaches 0. However, in this case, the application of L'Hôpital's Rule isn't necessary, as we could directly determine the derivative to be \( -\sin x \) at \( x = 0 \) and arrive at the conclusion that the limit is 0.

Understanding when and how to apply L'Hôpital's Rule is crucial, as its misuse can lead to incorrect conclusions. Hence, it's advised to first determine if the rule is applicable by checking for indeterminate forms and then proceed with its careful application.