Find lim_(theta rarr(pi)/(4))(cos(2theta))/(sqrt2cos(theta)-1). Choose 1 answer: (A) 2 (B) (1)/(2) (C) sqrt2 (D) The limit doesn't exist

Question

Find  lim_(theta rarr(pi)/(4))(cos(2theta))/(sqrt2cos(theta)-1). Choose 1 answer: (A) 2 (B)  (1)/(2) (C)  sqrt2 (D) The limit doesn't exist

Bytelearn · Accepted Answer

Identify Limit: Identify the limit that needs to be evaluated. We need to find the limit of the function (cos(2theta))/(sqrt2cos(theta)-1) as theta approaches pi/4.Check Direct Substitution: Substitute the value of theta into the function to check for direct substitution. If we substitute theta = pi/4 into the function, we get (cos(2 * pi/4))/(sqrt2 * cos(pi/4) - 1), which simplifies to (cos(pi/2))/(sqrt2 * (sqrt2/2) - 1). Since cos(pi/2) = 0, the numerator becomes 0. However, the denominator also becomes (1 - 1) = 0, which means we have an indeterminate form 0/0. Therefore, direct substitution does not work.Simplify Expression: Simplify the expression and try to eliminate the indeterminate form. We can use trigonometric identities to simplify the expression. The double angle formula for cosine is cos(2theta) = cos^2(theta) - sin^2(theta), which can also be written as cos(2theta) = 2cos^2(theta) - 1 since sin^2(theta) = 1 - cos^2(theta).Apply Double Angle Identity: Apply the double angle identity to the numerator. Using the identity from Step 3, we can rewrite the numerator as 2cos^2(theta) - 1. So the function becomes (2cos^2(theta) - 1)/(sqrt2cos(theta) - 1).Factor Numerator: Factor the numerator and look for common factors. The numerator 2cos^2(theta) - 1 can be factored as 2(cos^2(theta) - 1/2). We recognize that cos^2(theta) - 1/2 is the same as cos^2(theta) - (sqrt2/2)^2, which is a difference of squares and can be factored further.Factor Difference of Squares: Factor the difference of squares in the numerator. We can write cos^2(theta) - (sqrt2/2)^2 as (cos(theta) - sqrt2/2)(cos(theta) + sqrt2/2). Now the function is (2(cos(theta) - sqrt2/2)(cos(theta) + sqrt2/2))/(sqrt2cos(theta) - 1).Cancel Common Factor: Cancel out the common factor. We notice that (cos(theta) - sqrt2/2) is a common factor in both the numerator and the denominator. We can cancel this factor out, leaving us with (2(cos(theta) + sqrt2/2))/(sqrt2).Substitute and Simplify: Substitute theta = pi/4 into the simplified function. Now that we have simplified the function, we can substitute theta = pi/4 again. This gives us (2(cos(pi/4) + sqrt2/2))/(sqrt2). Since cos(pi/4) = sqrt2/2, the expression simplifies to (2(sqrt2/2 + sqrt2/2))/(sqrt2).Final Answer: Simplify the expression. The expression simplifies to (2(2sqrt2/2))/(sqrt2), which is (2sqrt2)/(sqrt2). The sqrt2 in the numerator and denominator cancel each other out, leaving us with the final answer of 2.

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

Full solution

More problems from Find derivatives of sine and cosine functions

Related topics