Find lim_(x rarr(pi)/(2))(cot^(2)(x))/(1-sin(x)) Choose 1 answer: (A) -1 (B) -(pi)/(2) (C) 2 (D) The limit doesn't exist

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Find  lim_(x rarr(pi)/(2))(cot^(2)(x))/(1-sin(x)) Choose 1 answer: (A) -1 (B)  -(pi)/(2) (C) 2 (D) The limit doesn't exist

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Understand Behavior of Function: First, let's understand the behavior of the function as x approaches π/2. The cotangent function, cot(x), is the reciprocal of the tangent function, tan(x), which means cot(x) = 1/tan(x). As x approaches π/2, tan(x) approaches infinity because the angle is approaching 90 degrees where the tangent function is not defined. Therefore, cot(x) approaches 0. However, we have cot^2(x) in the numerator, which means we need to consider the square of cot(x) as x approaches π/2.Evaluate Limit Indeterminate Form: Now let's look at the denominator, 1 - sin(x). As x approaches π/2, sin(x) approaches 1. This means that the denominator, 1 - sin(x), approaches 0. We have a situation where both the numerator and the denominator are approaching 0, which is an indeterminate form. This suggests that we may need to apply L'Hôpital's Rule to evaluate the limit.Simplify Expression: Before applying L'Hôpital's Rule, let's simplify the expression if possible. The expression cot^2(x) can be written as cos^2(x)/sin^2(x). So, the limit can be rewritten as: lim_(x -> π/2) (cos^2(x)/sin^2(x)) / (1 - sin(x)) This is equivalent to: lim_(x -> π/2) (cos^2(x)) / (sin^2(x) * (1 - sin(x)))Apply L'Hôpital's Rule: Now we can apply L'Hôpital's Rule because we have an indeterminate form of 0/0. L'Hôpital's Rule states that if the limit as x approaches a of f(x)/g(x) is 0/0 or ∞/∞, then the limit is the same as the limit of the derivatives of the numerator and the denominator. So we need to find the derivatives of cos^2(x) and sin^2(x) * (1 - sin(x)) with respect to x.Find Derivatives: The derivative of cos^2(x) with respect to x is -2cos(x)sin(x) using the chain rule. The derivative of sin^2(x) with respect to x is 2sin(x)cos(x), and the derivative of -sin^3(x) with respect to x is -3sin^2(x)cos(x). Therefore, the derivative of the denominator sin^2(x) * (1 - sin(x)) with respect to x is 2sin(x)cos(x) - 3sin^2(x)cos(x).Apply Derivatives to Rule: Now we can apply the derivatives to L'Hôpital's Rule: lim_(x -> π/2) (-2cos(x)sin(x)) / (2sin(x)cos(x) - 3sin^2(x)cos(x)) We can simplify this by canceling out common factors of cos(x) and sin(x) where possible, keeping in mind that as x approaches π/2, sin(x) approaches 1 and cos(x) approaches 0.Simplify and Substitute: After canceling out the common factors, we get: lim_(x -> π/2) (-2) / (2 - 3sin(x)) Now we can substitute x = π/2 into the simplified expression: (-2) / (2 - 3(1)) = (-2) / (2 - 3) = (-2) / (-1) = 2

Find $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cot ^{2}(x)}{1-\sin (x)}$ $\newline$ Choose $1$ answer: $\newline$ (A) $-1$ $\newline$ (B) $-\frac{\pi}{2}$ $\newline$ (C) $2$ $\newline$ (D) The limit doesn't exist

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Find $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cot ^{2}(x)}{1-\sin (x)}$ $\newline$ Choose $1$ answer: $\newline$ (A) $-1$ $\newline$ (B) $-\frac{\pi}{2}$ $\newline$ (C) $2$ $\newline$ (D) The limit doesn't exist