Problem 89 - Calculus Single Variable

Question

In each of Exercises \(89-92,\) investigate the given limit numerically and graphically. \(\lim _{x \rightarrow 0}(1-\cos (x)) / \sin (x)^{2}\)

Step-by-Step Solution

Answer

The limit is \( \frac{1}{2} \).

Step 1: Understand the limit expression

We are asked to find the limit \( \lim_{x \to 0} \frac{1 - \cos(x)}{\sin^2(x)} \). The expression involves trigonometric functions \( \cos(x) \) and \( \sin(x) \), and we need to evaluate this as \( x \) approaches 0.

Step 2: Substitute small values of x

To investigate this numerically, substitute small positive and negative values close to zero into the expression \( \frac{1 - \cos(x)}{\sin^2(x)} \), such as \( x = 0.1, 0.01, 0.001 \), etc., and observe the trend as \( x \) approaches 0.

Step 3: Apply trigonometric limits

Use known limits: \( \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \) and \( \lim_{x \to 0} \frac{1 - \cos(x)}{x^2} = \frac{1}{2} \). Rewriting the original expression, we have \( \frac{1 - \cos(x)}{\sin^2(x)} = \frac{\frac{1 - \cos(x)}{x^2}}{\left(\frac{\sin(x)}{x}\right)^2} \) and substituting these limits gives \( \frac{\frac{1}{2}}{1} = \frac{1}{2} \).

Step 4: Graphical approach

Graph the function \( y = \frac{1 - \cos(x)}{\sin^2(x)} \) near \( x = 0 \). Observe the behavior of the graph as \( x \) approaches zero from both the left and right.

Key Concepts

Numerical Investigation

Exploring the limit of a function numerically means substituting specific numbers close to the point of interest into the function. In this case, our focus is on \( \lim_{x \to 0} \frac{1 - \cos(x)}{\sin^2(x)} \). We take small values of \( x \), like 0.1, 0.01, and 0.001, and plug them into \( \frac{1 - \cos(x)}{\sin^2(x)} \).

By doing this, we take note of how the resulting output changes as \( x \) nears zero.

• For \( x = 0.1 \), calculate \( \frac{1 - \cos(0.1)}{\sin^2(0.1)} \) and observe the result.
• Similarly, perform the calculation for \( x = 0.01 \) and \( x = 0.001 \).

These computations help us see how the fraction behaves. We anticipate that as \( x \) approaches zero, the values will approach a specific number, helping verify the theoretical limit.

Graphical Approach

Drawing graphs can make abstract concepts more tangible, especially when dealing with limits. For the expression \( y = \frac{1 - \cos(x)}{\sin^2(x)} \), graphing can visually show what happens as \( x \) approaches zero.

When you plot this function on a graph, focus on the interval close to zero, such as between -0.1 and 0.1.

• Observe how the graph behaves near zero.
• Does the function seem to blow up, stabilize, or approach a horizontal line?

Such visual aids can reinforce the numerical findings and confirm that as \( x \to 0 \), the limit indeed approaches a particular value, helping us observe the behavior more intuitively.

Trigonometric Limits

To mathematically determine the limit \( \lim_{x \to 0} \frac{1 - \cos(x)}{\sin^2(x)} \), we use known trigonometric limits.

First, remember two important limits:

• \( \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \)
• \( \lim_{x \to 0} \frac{1 - \cos(x)}{x^2} = \frac{1}{2} \)

These formulas help simplify our function through substitution and rewriting. By expressing \( \frac{1 - \cos(x)}{\sin^2(x)} \) as \( \frac{\frac{1 - \cos(x)}{x^2}}{\left(\frac{\sin(x)}{x}\right)^2} \), we substitute the known limits to get \( \frac{\frac{1}{2}}{1} = \frac{1}{2} \), showing analytically that the limit is \( \frac{1}{2} \). This process emphasizes the elegance and precision of mathematical limits.

Trigonometric Functions

The trigonometric functions sine and cosine play a key role in understanding many mathematical concepts, including limits. Here, they are critical in determining \( \lim_{x \to 0} \frac{1 - \cos(x)}{\sin^2(x)} \).

Both \( \sin(x) \) and \( \cos(x) \) are fundamental periodic functions, featuring prominently in the principles of calculus and analytical geometry.

• \( \sin(x) \) describes the ratio of the opposite side to the hypotenuse in a right triangle.
• \( \cos(x) \) describes the ratio of the adjacent side to the hypotenuse.

Values often oscillate between -1 and 1, influencing the behavior of the overall limit. Understanding these trigonometric basics helps interpret how tiny tweaks near zero affect both \( \cos(x) \) and \( \sin(x) \), critical in finding precise limit outcomes.