Question

Evaluate the following limits. $$\lim _{x \rightarrow 0} \frac{\cos x-1}{\sin ^{2} x}$$

Short answer

Question: Determine the limit of the given expression as x approaches 0: $$\lim _{x \rightarrow 0} \frac{\cos x-1}{\sin ^{2} x}$$ Answer: The limit of the given expression as x approaches 0 is -1.

Step-by-step solution

Rewrite the given limit using the trigonometric identity

Using the identity \(\cos^2 x + \sin^2 x = 1\), we can rewrite the given expression as: $$\frac{(1 - \sin^2 x) - 1}{\sin^2 x}$$ This simplifies to: $$\frac{-\sin^2 x}{\sin^2 x}$$

Simplify the expression

We can now simplify the expression by canceling the \(\sin^2 x\) term in the numerator and denominator. $$\frac{-\sin^2 x}{\sin^2 x} = -1$$

Evaluate the limit

Since the expression simplified to a constant of -1, the limit will also be -1, as x approaches 0. Therefore, the result is: $$\lim _{x \rightarrow 0} \frac{\cos x-1}{\sin ^{2} x} = -1$$

Key concepts

Trigonometric identities

Trigonometric identities are fundamental tools in calculus, especially when evaluating limits. These identities are equations that hold true for all angles and help simplify expressions. One essential identity is

• \( \cos^2 x + \sin^2 x = 1 \), which expresses the Pythagorean relationship in trigonometry. This identity is incredibly useful for converting expressions involving trigonometric functions.
• Another important identity is: \( 1 - \cos x = 2 \sin^2 \left( \frac{x}{2} \right) \), which can further be used to simplify complex trigonometric limits.

By rearranging these identities, we can break down complex expressions into manageable forms. For instance, in the provided exercise, the expression \( \cos x - 1 \) is transformed using the trigonometric identity to simplify the overall limit evaluation process. Understanding these identities not only helps in solving problems efficiently but also unveils the underlying structure of trigonometric functions.

Limit simplification

Simplifying limits is a crucial aspect of calculus, and it often involves algebraic manipulation or utilizing specific trigonometric identities. The goal is to transform complex expressions into simpler forms that are easier to evaluate.
When dealing with trigonometric functions, it's common to encounter indeterminate forms like \( \frac{0}{0} \). In such cases, rewriting the terms or substituting equivalent expressions can eliminate these indeterminate conditions.
In the original exercise, we start with the fraction \( \frac{\cos x - 1}{\sin^2 x} \). By applying the identity \( \cos^2 x + \sin^2 x = 1 \), it is simplified to \( \frac{-\sin^2 x}{\sin^2 x} \).

• This equation can then be reduced by canceling out the common term \( \sin^2 x \) in both the numerator and the denominator, leaving a manageable form of \(-1\).

Such simplification removes complexities and reveals constant expressions that make limits much more straightforward to evaluate. Becoming adept at these techniques allows one to more efficiently understand and solve limit problems.

Trigonometric limits

Trigonometric limits are a specialized subset of limits dealing with trigonometric functions as their variables approach a certain value. These types of limits often require a good grasp of basic limit properties, as well as familiarity with trigonometric identities.
A common scenario involves calculating limits as the angle approaches zero. Such expressions can often lead to indeterminate forms, necessitating tools like L'Hopital's Rule or identities to simplify them.

• In the exercise provided, the limit \( \lim_{x \rightarrow 0} \frac{\cos x - 1}{\sin^2 x} \) evaluates a situation where both the numerator and denominator tend towards zero, classifying it as an indeterminate form.
• However, by rewriting and simplifying the expression (as demonstrated with trigonometric identities), we ended up with \(-1\), which means that as \(x\) approaches zero, the value of the limit stabilizes to \(-1\).

Mastering trigonometric limits involves understanding these transformations and applications, often relying on both algebraic and trigonometric properties to navigate around indeterminate situations.