Problem 8 - Thomas Calculus

Question

Use 1'Hopital's Rule to find the limits in Exercises \(7-26\). $$ \lim _{x \rightarrow \pi / 2} \frac{2 x-\pi}{\cos x} $$

Step-by-Step Solution

Answer

The limit is -2.

Step 1: Determine the Indeterminate Form

Substitute the limit point \(x = \frac{\pi}{2}\) into the function \(\frac{2x - \pi}{\cos x}\) to see if it results in an indeterminate form. Substituting gives: \(\frac{2(\pi/2) - \pi}{\cos(\pi/2)} = \frac{\pi - \pi}{0} = \frac{0}{0}\). This is an indeterminate form, so we can use L'Hopital's Rule.

Step 2: Apply L'Hopital's Rule

L'Hopital's Rule states that for functions \(f(x)\) and \(g(x)\) that are differentiable, if \(\lim_{x\to c} f(x) = \lim_{x\to c} g(x) = 0\) or \(\pm \infty\), then \(\lim_{x\to c} \frac{f(x)}{g(x)} = \lim_{x\to c} \frac{f'(x)}{g'(x)}\). For our function, \(f(x) = 2x - \pi\) and \(g(x) = \cos x\).

Step 3: Differentiate the Numerator and Denominator

Find the derivative of the numerator \(f(x) = 2x - \pi\), which is \(f'(x) = 2\). Then find the derivative of the denominator \(g(x) = \cos x\), which is \(g'(x) = -\sin x\).

Step 4: Evaluate the Limit with the Derivatives

Now compute the limit using the derivatives: \(\lim_{x\to \pi/2} \frac{f'(x)}{g'(x)} = \lim_{x\to \pi/2} \frac{2}{-\sin x}\). Substitute \(x = \frac{\pi}{2}\), so \(-\sin\left(\frac{\pi}{2}\right) = -1\). Thus, \(\frac{2}{-1} = -2\).

Key Concepts

Indeterminate Forms

Indeterminate forms arise when evaluating the limit of a function results in an uncertain outcome. In mathematical terms, these forms generally come in different types such as \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0 \, \cdot \, \infty\), and others. In the context of L'Hopital's Rule, the most common are \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\).
This means that at the point where you're evaluating a limit, direct substitution into the function doesn't provide any useful information about the limit.

To identify an indeterminate form:

• Directly substitute the limit value into the original function \(\frac{2x - \pi}{\cos x}\).
• If you end up with \(\frac{0}{0}\) or any other indeterminate form after substitution, L’Hôpital's Rule can be used to resolve it.

Once an indeterminate form is confirmed, this is your cue to proceed with differentiation to arrive at a solution using L'Hopital's Rule.

Differentiation

Differentiation is a crucial concept in calculus, allowing us to determine how functions change. It involves finding the derivative of a function, which represents the slope of the function at any point.
The process of differentiation is applied to the numerator and denominator of the function in question. For L'Hopital's Rule:

• First, find the derivative of the numerator \(f(x) = 2x - \pi\), which is \(f'(x) = 2\).
• Next, find the derivative of the denominator \(g(x) = \cos x\), which is \(g'(x) = -\sin x\).

Differentiating provides you with new functions to evaluate the limit.
This step is fundamental because it transforms an indeterminate form into one that is more easily evaluated by substituting the limit value into \(\frac{f'(x)}{g'(x)}\).
By understanding the behavior of the original functions, we can examine how the transformed function behaves, setting us up to apply the next step: limit evaluation.

Limit Evaluation

Limit evaluation is the final step in settling on the actual value at which the function approaches as \(x\) approaches a certain point. After having differentials for both the numerator and the denominator, we are set to calculate the limit itself.

For situations involving L'Hopital's Rule:

• Determine the limit of the new function \(\lim_{x\to c} \frac{f'(x)}{g'(x)}\). In this exercise, it's \(\lim_{x\to \pi/2} \frac{2}{-\sin x}\).
• Substitute \(x = \frac{\pi}{2}\) to find the exact value \(-\sin\left(\frac{\pi}{2}\right) = -1\).
• Calculate \(\frac{2}{-1} = -2\).

This result shows that as \(x\) approaches \(\frac{\pi}{2}\), the original function approaches \(-2\).
Understanding how to evaluate limits is key to unraveling the behavior of functions at specific points, helping to predict how they behave near certain values, and providing a clearer insight into their structure.