Problem 36 - Precalculus with Limits

Question

In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically. $$\lim_{x \to \pi/2} \dfrac{\cos\ x - 1}{\sin\ x}$$

Step-by-Step Solution

Answer

The limit of the given function as \(x\) approaches \(\pi/2\) does not exist.

Step 1: Identify the indeterminate form

As \(x\) is approaching \(\pi/2\), \(\cos x - 1\) approaches \(0\) and \(\sin x\) approaches \(1\). Hence, the function might have the form 0/0 which is indeterminate.

Step 2: Apply L'Hospital's rule

Given that we have an indeterminate form of type \(0/0\), we can apply L’Hospital's rule. According to L'Hospital's rule, if the limit of a function results in an indeterminate form like \(0/0\) or \(\infty/\infty\), we can take the derivatives of the numerator and the denominator separately and then try taking the limit again. Therefore, we substitute \(\frac{\cos x - 1}{\sin x}\) with \(\frac{d/dx(\cos x - 1)}{d/dx(\sin x)}\), which simplifies to \(\frac{-\sin x}{\cos x}\) or \(-\tan x\).

Step 3: Compute the Limit

After using L'Hospital's rule, the limit is now \(\lim_{x \to \pi/2} -\tan x\). The \(\tan\) function approaches \(\infty\) as \(x\) approaches \(\pi/2\) from the left and \(-\infty\) as \(x\) approaches \(\pi/2\) from the right. Hence, this limit is undefined since it does not approach a particular finite number.

Step 4: Verify the solution graphically

To double check the calculations, graph \(\frac{\cos x - 1}{\sin x}\) and see how it behaves around \(x = \pi/2\). By doing so, it is noticed that the function indeed approaches negative infinity from the right and positive infinity from the left, thus confirming our earlier analysis that the limit does not exist.

Key Concepts

L'Hospital's Rule

L'Hospital's Rule is a very handy tool in calculus when dealing with difficulties in finding limits, particularly when they result in indeterminate forms like \(0/0\) or \(\infty/\infty\). It helps us simplify these problems by allowing us to take the derivative of the numerator and the denominator separately. This is only applicable if the original limit results in one of those indeterminate forms.
When you encounter such a limit, you first confirm it's an indeterminate form. Then, apply L'Hospital's Rule by differentiating the top and bottom separately and re-evaluate the limit. If using L'Hospital's Rule doesn't resolve the indeterminate form, you may apply the rule again, assuming the conditions are still met.
In our example, applying the rule to \(\lim_{x \to \pi/2} \frac{\cos x - 1}{\sin x}\) involved differentiating to get \(-\tan x\). This step turned the original indeterminate form into something easier to analyze, even if the limit didn't exist in the end.

Indeterminate Forms

When calculating limits, sometimes you'll encounter expressions that aren't initially straightforward, such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). These are called indeterminate forms because they don't immediately provide information about the limit’s value.
An indeterminate form occurs in our example where \(\cos x - 1\) approaches zero and \(\sin x\) approaches one as \(x\) approaches \(\pi/2\). Despite this, the overall fraction remains in a tricky \(0/0\) form, indicating that L'Hospital's Rule might be necessary.
Indeterminate forms beg further investigation; that’s why we have tools like L'Hospital's Rule. They signal that usual limit evaluation tactics, such as direct substitution, don't work and require different methods to find the answer or prove the limit doesn’t exist.

Trigonometric Limits

Understanding trigonometric limits is key to solving problems where angles are involved. Many trigonometric limits often rely on recognizing patterns or remembering key limits, like \(\lim_{x \to 0} \frac{\sin x}{x} = 1\).
In the exercise given, the limit involves trigonometric functions \(\cos x\) and \(\sin x\). Recognizing how these functions behave as \(x\) approaches specific values helps you predict the limit’s behavior. For example, as \(x\) approaches \(\pi/2\):

• \(\cos x\) approaches zero,
• \(\sin x\) approaches one.

This behavior is unproblematic on its own but when put into a fraction like our example, further steps like using L'Hospital's rule are necessary to understand the exact behavior near the limit point.