Problem 16 - Essential Calculus Early Transcendentals

Question

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 0} \frac{\cos m x-\cos n x}{x^{2}}$$

Step-by-Step Solution

Answer

\(\frac{n^2 - m^2}{2}\)

Step 1: Check Indeterminate Form

First, substitute \(x = 0\) in the given limit expression \(\frac{\cos(m x) - \cos(n x)}{x^2}\). We have \(\cos(m \cdot 0) - \cos(n \cdot 0) = 1 - 1 = 0\), making the whole expression \(\frac{0}{0}\) which is an indeterminate form.

Step 2: Verify L'Hospital's Rule Applicability

Since the expression is of \(\frac{0}{0}\) indeterminate form, L'Hospital's Rule can be applied. This rule allows us to find the limit of a function by differentiating the numerator and the denominator separately.

Step 3: Differentiate Numerator and Denominator

Differentiate the numerator \(\cos(m x) - \cos(n x)\). The derivative is \(-m \sin(m x) + n \sin(n x)\).The derivative of the denominator \(x^2\) is \(2x\).

Step 4: Apply L'Hospital's Rule

Apply L'Hospital's Rule:\[\lim _{x \rightarrow 0} \frac{-m \sin(m x) + n \sin(n x)}{2x}\]

Step 5: Assess Resulting Indeterminate Form

Substitute \(x = 0\) again in this new expression:\(-m \sin(0) + n \sin(0) = 0\).So, we have another \(\frac{0}{0}\) form, indicating that L'Hospital's Rule applies again.

Step 6: Differentiate Again

Differentiate again:- Numerator: The derivative of \(-m \sin(m x) + n \sin(n x)\) is \(-m^2 \cos(m x) + n^2 \cos(n x)\).- Denominator: The derivative of \(2x\) is \(2\).

Step 7: Evaluate the Limit

Now, find the limit:\[\lim _{x \rightarrow 0} \frac{-m^2 \cos(m x) + n^2 \cos(n x)}{2}\]Substitute \(x = 0\):We get \(-m^2(1) + n^2(1) = n^2 - m^2\).Therefore, the limit is \(\frac{n^2 - m^2}{2}\).

Key Concepts

Indeterminate Forms

In calculus, understanding indeterminate forms is crucial for dealing with limits that are not straightforward. An indeterminate form happens when direct substitution into a limit expression yields a form such as \(\frac{0}{0}\), \(\infty - \infty\), or \(\frac{\infty}{\infty}\). These forms do not provide clear information about the limit, thus requiring advanced methods to solve them.

For instance, in our exercise, substituting \(x = 0\) into the expression \(\frac{\cos(m x) - \cos(n x)}{x^2}\) results in \(\frac{0}{0}\). This is an indeterminate form, which indicates that we cannot determine the limit by direct substitution alone.

• Indeterminate forms prompt us to utilize techniques such as L'Hospital's Rule to resolve the limit.
• They occur because the behavior near the limit point is not straightforward, and simple evaluation does not suffice.

By recognizing indeterminate forms, we can apply appropriate calculus tools to find precise results.

Differentiation

Differentiation is a fundamental process in calculus that involves finding the derivative of a function. The derivative represents the rate at which a function changes at any given point. It's essential when working with limits and indeterminate forms, especially when applying L'Hospital's Rule.

In the exercise, differentiation plays a key role when we encounter the indeterminate form \(\frac{0}{0}\). By differentiating the numerator and the denominator separately, we transform the original limit problem into a simpler one.

• The derivative of the numerator \(\cos(m x) - \cos(n x)\) is \(-m \sin(m x) + n \sin(n x)\).
• The derivative of the denominator \(x^2\) is \(2x\).

After performing these differentiations, L'Hospital's Rule is applied again to further resolve the limit. Differentiation simplifies the problem and takes us closer to finding a concrete limit value.

Limits in Calculus

Limits are a cornerstone of calculus, representing the value that a function approaches as the input approaches a particular point. They are vital for understanding the behavior of functions, especially near points of indeterminate forms and discontinuities.

In this exercise, we are tasked with finding the limit of the given function as \(x\) approaches 0. Initiating with the expression \(\frac{\cos(m x) - \cos(n x)}{x^2}\), direct substitution leads to the indeterminate form \(\frac{0}{0}\). This tells us that the behavior near \(x = 0\) is more complex, necessitating further analysis, such as the use of L'Hospital's Rule.

• Limits help in identifying how functions behave near troublesome points where typical computation fails.
• Understanding limits is key to solving indeterminate forms and fully comprehending the implications of L'Hospital's Rule.

The final limit calculated provides critical insights into the behavior of the function, revealing that the limit is \(\frac{n^2 - m^2}{2}\). Knowing how to calculate limits equips you with powerful analytical tools in calculus.