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

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Calculus

Find derivatives of sine and cosine functions

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Q. 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

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