Question

Use the trigonometric identity \(\cos ^{2} x+\sin ^{2} x=1\) to prove that \(\lim _{x \rightarrow 0} \frac{\cos x-1}{x}=0 .\) (Hint: Begin by multiplying the numerator and denominator by \(\cos x+1 .)\)

Short answer

Answer: The limit of the given expression as x approaches 0 is -1/2.

Step-by-step solution

Multiply numerator and denominator by \(\cos x + 1\)

As given in the hint, we begin by multiplying the numerator and denominator of the expression by \(\cos x + 1\): \(\lim _{x \rightarrow 0} \frac{\cos x-1}{x} \cdot \frac{\cos x + 1}{\cos x + 1} = \lim _{x \rightarrow 0} \frac{(\cos x-1)(\cos x+1)}{x(\cos x + 1)}\)

Apply the difference of squares

In the numerator, we can recognize the difference of squares. So, let's apply this formula: \((\cos x - 1)(\cos x + 1) = \cos ^2 x - 1^2\) Now, substitute this expression in the limit: \(\lim _{x \rightarrow 0} \frac{\cos^2 x - 1^2}{x(\cos x + 1)}\)

Use trigonometric identity

Recall the given trigonometric identity, \(\cos ^{2} x+\sin ^{2} x=1\). We can rearrange this identity to express \(\cos^2 x\) as \(1 - \sin^2 x\): \(\cos^2 x = 1 - \sin^2 x\) Substitute this expression in the limit: \(\lim _{x \rightarrow 0} \frac{(1 - \sin^2 x) - 1^2}{x(\cos x + 1)}\) Simplify the expression: \(\lim _{x \rightarrow 0} \frac{-\sin^2 x}{x(\cos x + 1)}\)

Apply limit laws

Now, we'll apply the limit laws to solve this expression. To evaluate the limit successfully, rewrite it as: \(\lim _{x \rightarrow 0} \frac{-\sin^2 x}{x} \cdot \frac{1}{\cos x + 1}\) Now, we can separately evaluate both limits: \(\lim _{x \rightarrow 0} \frac{-\sin^2 x}{x} \cdot \lim _{x \rightarrow 0} \frac{1}{\cos x + 1}\) For the first limit, we can use the fact that \(\lim _{x \rightarrow 0} \frac{\sin x}{x} = 1\), so: \(\lim _{x \rightarrow 0} \frac{-\sin^2 x}{x} = -(\lim _{x \rightarrow 0} \frac{\sin x}{\sqrt{x}})(\lim _{x \rightarrow 0} \frac{\sin x}{\sqrt{x}}) = -1\cdot1\) For the second limit, when \(x\) approaches \(0\), \(\cos x\) approaches \(1\): \(\lim _{x \rightarrow 0} \frac{1}{\cos x + 1} = \frac{1}{1 + 1} = \frac{1}{2}\) Now, multiply both limits: \((-1) \cdot \frac{1}{2} = -\frac{1}{2}\) So, the solution to the given exercise is: \(\lim_{x \rightarrow 0} \frac{\cos x-1}{x} = -\frac{1}{2}\)

Key concepts

Trigonometric Identity

Trigonometric identities play a vital role in simplifying expressions in both trigonometry and calculus. One of the most fundamental identities is the Pythagorean identity: \[ \cos ^{2} x + \sin ^{2} x = 1 \.\] It expresses the inherent relationship between the sine and cosine functions and states that, for any angle \( x \), the square of the sine of the angle plus the square of the cosine of the angle always equals one.

Using this identity can greatly simplify complex expressions that involve trigonometric functions. As in the case of our exercise, the Pythagorean identity allows us to represent \( \cos^2 x \) in terms of \( \sin^2 x \), which paves the way for further simplification when finding limits. When faced with a trigonometric expression within a limit problem, always check if applying an identity can help to simplify the trouble spots.

Difference of Squares

The difference of squares is a commonly used algebraic technique that facilitates the factorization of certain polynomials. The formula can be stated as: \[ a^2 - b^2 = (a + b)(a - b) \.\] In our textbook exercise, the difference of squares was applied to the expression \( \cos x - 1 \), treated as \( (\cos x)^2 - (1)^2 \).

Utilizing this formula transformed the limit expression into a product of two terms, one of which canceled out the plus one in the denominator, thus reducing the complexity of the original limit. A key advantage of recognizing and employing the difference of squares method is that it quickly breaks down expressions into simpler parts, making the evaluation of limits more straightforward.

Limit Laws

Limit laws are a set of rules that provide the foundation for calculating limits in calculus. They are essential tools for analyzing the behavior of functions as the input value \( x \) approaches a specific point. Some basic limit laws include:

• The Sum Law: The limit of the sum is the sum of the limits.
• The Product Law: The limit of a product is the product of the limits.
• The Quotient Law: The limit of a quotient is the quotient of the limits (provided the limit of the denominator isn’t zero).

By applying limit laws, complex expressions can often be broken down into simpler parts that can be evaluated separately. In the exercise, the limit of a product was used to separate the given limit into two individual limits that could be calculated easily. It's crucial to remember that limit laws can only be applied when the limits exist. Therefore, always ensure that the limits you're evaluating as part of these laws are indeed definable before applying them.