Find lim_(x rarr(pi)/(4))(cos(2x))/(cos(x)-sin(x)). Choose 1 answer: (A) sqrt2 (B) 2 (C) 4 (D) The limit doesn't exist

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Find  lim_(x rarr(pi)/(4))(cos(2x))/(cos(x)-sin(x)). Choose 1 answer: (A)  sqrt2 (B) 2 (C) 4 (D) The limit doesn't exist

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Identify Limit: Identify the limit that needs to be evaluated. We need to find the limit of the function (cos(2x))/(cos(x)-sin(x)) as x approaches π/4.Substitute Value: Substitute the value of x into the function to see if the limit can be directly evaluated. lim_(x -> π/4)(cos(2x))/(cos(x)-sin(x)) = (cos(2(π/4)))/(cos(π/4)-sin(π/4))Simplify Expression: Simplify the expression by using trigonometric identities and the known values of cos(π/4) and sin(π/4). cos(2(π/4)) = cos(π/2) = 0 cos(π/4) = sin(π/4) = sqrt(2)/2 So, the expression becomes 0/(sqrt(2)/2 - sqrt(2)/2)Evaluate Denominator: Evaluate the denominator and check for any indeterminate forms. The denominator is sqrt(2)/2 - sqrt(2)/2 = 0 Since the numerator is also 0, we have an indeterminate form of 0/0.Apply L'Hôpital's Rule: 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 f'(x)/g'(x) as x approaches a, provided that the latter limit exists.Find Derivatives: Find the derivatives of the numerator and the denominator. The derivative of cos(2x) with respect to x is -2sin(2x). The derivative of cos(x) - sin(x) with respect to x is -sin(x) - cos(x).Evaluate New Limit: Evaluate the new limit using the derivatives. lim_(x -> π/4)(-2sin(2x))/(-sin(x) - cos(x)) Substitute x = π/4 into the derivatives: -2sin(2(π/4)) = -2sin(π/2) = -2 -sin(π/4) - cos(π/4) = -sqrt(2)/2 - sqrt(2)/2 = -sqrt(2)Calculate Final Limit: Calculate the limit using the values from Step 7. lim_(x -> π/4)(-2sin(2x))/(-sin(x) - cos(x)) = -2/(-sqrt(2)) = 2/sqrt(2) = sqrt(2)Choose Correct Answer: Choose the correct answer based on the calculation. The limit is sqrt(2), which corresponds to answer choice (A).

Find $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\cos (2 x)}{\cos (x)-\sin (x)}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\sqrt{2}$ $\newline$ (B) $2$ $\newline$ (C) $4$ $\newline$ (D) The limit doesn't exist

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Find $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\cos (2 x)}{\cos (x)-\sin (x)}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\sqrt{2}$ $\newline$ (B) $2$ $\newline$ (C) $4$ $\newline$ (D) The limit doesn't exist