Problem 51 - A Graphical Approach to Precalculus with Limits

Question

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 0} \frac{\cos x-1}{x}\)

Step-by-Step Solution

Answer

The limit exists and is 0.

Step 1: Understand the problem

We need to find the limit \( \lim _{x \rightarrow 0} \frac{\cos x - 1}{x} \). This involves figuring out the behavior of the function as \( x \) approaches 0.

Step 2: Investigate using a table

Create a table of \( x \) values approaching 0 from both the negative and positive sides, compute \( \frac{\cos x - 1}{x} \) for each and observe the pattern.| \( x \) | \( \cos x - 1 \) | \( \frac{\cos x - 1}{x} \) ||--------------|----------------|------------------------------|| -0.1 | -0.00499 | 0.0499 || -0.01 | -0.00004999 | 0.004999 || -0.001 | -0.0000004999 | 0.0004999 || 0.001 | -0.0000004999 | -0.0004999 || 0.01 | -0.00004999 | -0.004999 || 0.1 | -0.00499 | -0.0499 |

Step 3: Analyze the table results

From the table, as \( x \) approaches 0 from both sides, \( \frac{\cos x - 1}{x} \) seems to approach 0. Thus, the limit appears to exist, approaching 0.

Step 4: Confirm with L'Hôpital's Rule

Since we have an indeterminate form \( \frac{0}{0} \), apply L'Hôpital's Rule:\[ \lim_{x \to 0} \frac{\cos x - 1}{x} = \lim_{x \to 0} \frac{-\sin x}{1} = 0 \]Thus confirming our observation from the table.

Key Concepts

L'Hôpital's Rule

When you encounter a limit problem like \( \lim _{x \rightarrow 0} \frac{\cos x - 1}{x} \), you might run into what is known as an indeterminate form—more on that later. L'Hôpital's Rule is a handy tool in such cases. It helps to evaluate limits like these by differentiating the numerator and the denominator until a clear limit emerges. Here’s how it works: if you have a limit \( \lim_{x \to c} \frac{f(x)}{g(x)} \) resulting in \( \frac{0}{0} \) or \( \frac{\pm \infty}{\pm \infty} \), you can differentiate the top and the bottom:

• Derive the numerator \( f'(x) \) and denominator \( g'(x) \).
• Evaluate the limit of \( \lim_{x \to c} \frac{f'(x)}{g'(x)} \).

Apply this repeatedly if necessary. For our problem, differentiating \( \cos x - 1 \) gives \( -\sin x \) and differentiating \( x \) gives \( 1 \). Thus, the limit \( \lim_{x \to 0} \frac{-\sin x}{1} = 0 \) confirms that the original limit evaluates to 0.
Using L'Hôpital's Rule simplifies what seems like complex calculus into a series of straightforward steps.

Trigonometric Functions

Trigonometric functions like \( \cos x \) and \( \sin x \) are fundamental in calculus and help us understand periodic phenomena. When evaluating limits involving these functions, it’s vital to grasp their behavior around critical points.

\( \cos x \) approximates 1 when \( x \) is close to 0.

\( \sin x \) approaches 0 when \( x \) approaches 0.

In our specific scenario, we deal with the expression \( \cos x - 1 \). When plugged into a limit problem as \( \frac{\cos x - 1}{x} \), noticing how each component behaves as \( x \) nears 0 is essential.
The function \( \cos x \) changes from its maximum and the subtraction by 1 results in values that tend towards zero. Understanding this behavior shapes how we analyze the limit: the cosine wave's gentle slope around zero leads to a limit of 0 when combined with L'Hôpital's Rule.

Indeterminate Forms

Indeterminate forms occur when a straightforward evaluation of a limit doesn't yield a specific value. Forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) are classified as indeterminate because their nature doesn't directly point to a definite result. Instead, we need additional strategies to resolve them. Here’s how you can handle indeterminate forms:

• Try simplifying algebraically.
• Use techniques such as factoring, rationalizing, or expanding.
• If still indeterminate, apply L'Hôpital's Rule if applicable.

In the case of \( \lim _{x \rightarrow 0} \frac{\cos x - 1}{x} \), initially substituting \( x = 0 \) gives \( \frac{0}{0} \), an indeterminate form.
This tells us we need a deeper analysis or a rule like L'Hôpital’s to solve it. Indeterminate forms compel us to look beyond basic algebra and leverage calculus tools to find solutions.
Understanding these forms and how to manage them is a crucial skill, turning challenging limits into manageable calculations.