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.