Question

, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{x \rightarrow 0} \frac{(1-\cos x)^{2}}{x^{2}} $$

Short answer

The limit is 0.

Step-by-step solution

Understand the Problem

We need to find the limit of the function \( f(x) = \frac{(1 - \cos x)^2}{x^2} \) as \( x \) approaches 0.

Use Trigonometric Identities

Recall the trigonometric identity \( 1 - \cos x = 2 \sin^2\left(\frac{x}{2}\right) \). Substitute this into the function to rewrite it as \( \frac{\left(2\sin^2(\frac{x}{2})\right)^2}{x^2} \).

Simplify the Expression

The expression becomes \( \frac{4\sin^4(\frac{x}{2})}{x^2} \). Observe the inner function \( \sin\left(\frac{x}{2}\right) \).

Express Using Limits

Consider \( u = \frac{x}{2} \), then as \( x \rightarrow 0, u \rightarrow 0 \). Rewrite the limit as \( \lim_{u \rightarrow 0} \frac{4 \sin^4(u)}{4u^2} \) because \( x = 2u \), so \( x^2 = 4u^2 \).

Apply Limit Properties

Simplify the expression further to \( \lim_{u \rightarrow 0} \sin^2(u) \left(\frac{\sin(u)}{u}\right)^2 \). We know the limit \( \lim_{u \rightarrow 0} \frac{\sin(u)}{u} = 1 \).

Compute the Limit

Using \( \lim_{u \rightarrow 0} \frac{\sin(u)}{u} = 1 \), we have \( \lim_{u \rightarrow 0} \sin^2(u) \cdot 1^2 = \lim_{u \rightarrow 0} \sin^2(u) = 0 \) because \( \sin(0) = 0 \).

Verify with Graph and Calculator

Use a graphing calculator to plot the function near \( x = 0 \). The graph should approach 0, confirming the analytical result.

Key concepts

Trigonometric Limits

Trigonometric limits often appear complex, yet they can become manageable using known identities and properties. In this exercise, we encounter the limit \( \lim_{x \rightarrow 0} \frac{(1 - \cos x)^2}{x^2} \). Trigonometric identities are valuable tools to simplify such expressions. Here, the identity \( 1 - \cos x = 2 \sin^2\left(\frac{x}{2}\right) \) is used to transform the function.
Substituting \( 1 - \cos x \) with its equivalent, \( 2 \sin^2\left(\frac{x}{2}\right) \), simplifies the expression, making it easier to evaluate the limit. As a result, the function becomes \( \frac{4 \sin^4(\frac{x}{2})}{x^2} \) before proceeding to further simplifications. This approach reveals the hidden simplicity in what initially appears to be a complex limit problem.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful technique for evaluating indeterminate forms such as \( \frac{0}{0} \), which can frequently arise in limit problems. Although not directly applied in this exercise, it's worthy of mention for recognizing such forms within using alternative methods.
In scenarios where direct substitution of the limit into the equation results in an indeterminate form, L'Hôpital's Rule allows us to differentiate the numerator and denominator separately and then recompute the limit. In this particular problem, however, trigonometric identities and limit simplification suffice, showcasing that sometimes knowing multiple methods can guide you to choose the simplest path to the solution.

Graphing Calculators

Graphing calculators are invaluable tools when dealing with functions, particularly those involving trigonometric limits. They allow you to visualize the behavior of a function as it approaches a limit, offering an intuitive confirmation of analytical work.

• Plot the function \( \frac{(1 - \cos x)^2}{x^2} \) around \( x = 0 \) using a graphing calculator.
• Observe the trend of the graph as \( x \) nears zero — it should steadily approach a y-value of 0, consistent with our calculated analytical limit.

Graphing such functions provides strong, visual confirmation of the analytic steps taken to solve the problem. It helps reinforce the understanding of the limit behavior in real-time.

Limit Simplification Techniques

Simplifying limits requires a deep understanding of algebraic manipulation and mathematical identities. In this case, substituting \(1 - \cos x\) with \(2 \sin^2\left(\frac{x}{2}\right)\) simplifies what seems like a complicated expression. By reformulating:- Convert \( x \) into \( 2u \) such that \( u = \frac{x}{2} \), leading to \( x^2 = 4u^2 \).- Transform the limit from \( \lim_{x \rightarrow 0} \) to \( \lim_{u \rightarrow 0} \), which simplifies calculations.- Utilize known limits like \( \lim_{u \rightarrow 0} \frac{\sin(u)}{u} = 1 \).Applying these steps reduces the complex problem into recognizable parts, showcasing that persistence and insight into the properties of trigonometric functions can yield clear solutions. Such techniques are fundamental for finding limits without resorting to more complicated procedures.