Evaluate each limit. 


lim x→0 cos x−1 / sin^2x

Verified step by step guidance

1. Step 1: Recognize that the limit involves an indeterminate form 0/0 as x approaches 0, which suggests the use of L'Hôpital's Rule. L'Hôpital's Rule states that if the limit of f(x)/g(x) as x approaches a value results in an indeterminate form, the limit can be evaluated as the limit of f'(x)/g'(x).
2. Step 2: Differentiate the numerator and the denominator separately. The derivative of the numerator, cos(x) - 1, is -sin(x). The derivative of the denominator, sin^2(x), is 2sin(x)cos(x) using the chain rule.
3. Step 3: Apply L'Hôpital's Rule by taking the limit of the new fraction formed by the derivatives: lim x→0 [-sin(x)] / [2sin(x)cos(x)].
4. Step 4: Simplify the expression. Notice that the sin(x) terms in the numerator and denominator can be canceled out, resulting in the limit of -1 / [2cos(x)] as x approaches 0.
5. Step 5: Evaluate the simplified limit by substituting x = 0 into the expression -1 / [2cos(x)]. Since cos(0) = 1, the expression simplifies further.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's behavior near points of interest, including points where the function may not be explicitly defined. Evaluating limits is essential for determining continuity, derivatives, and integrals.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) as x approaches a point yields an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule simplifies the process of finding limits in complex expressions.

Trigonometric Limits

Trigonometric limits involve the evaluation of limits that include trigonometric functions, such as sine and cosine. These limits often require special techniques or known limit values, such as lim x→0 (sin x)/x = 1. Understanding the behavior of trigonometric functions near specific points is crucial for solving problems involving limits in calculus.