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Find lim_(theta rarr-(pi)/(4))(1+sqrt2sin(theta))/(cos(2theta)). Choose 1 answer: (A) 0 (B) (1)/(2) (C) (1)/(4) (D) The limit doesn't exist

Find lim⁡θ→−π41+2sin⁡(θ)cos⁡(2θ) \lim _{\theta \rightarrow-\frac{\pi}{4}} \frac{1+\sqrt{2} \sin (\theta)}{\cos (2 \theta)} .\newlineChoose 11 answer:\newline(A) 00\newline(B) 12 \frac{1}{2} \newline(C) 14 \frac{1}{4} \newline(D) The limit doesn't exist

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Full solution

Q. Find lim⁡θ→−π41+2sin⁡(θ)cos⁡(2θ) \lim _{\theta \rightarrow-\frac{\pi}{4}} \frac{1+\sqrt{2} \sin (\theta)}{\cos (2 \theta)} .\newlineChoose 11 answer:\newline(A) 00\newline(B) 12 \frac{1}{2} \newline(C) 14 \frac{1}{4} \newline(D) The limit doesn't exist
  1. Simplify Expression: We are asked to find the limit of the function (1+2sin⁡(θ))/(cos⁡(2θ))(1+\sqrt{2}\sin(\theta))/(\cos(2\theta)) as θ\theta approaches −π4-\frac{\pi}{4}. To do this, we will first simplify the expression if possible and then substitute the value of θ\theta into the simplified expression.
  2. Apply Double Angle Formula: Let's recall the double angle formula for cosine: cos⁡(2θ)=cos⁡2(θ)−sin⁡2(θ)\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta). We can also use the Pythagorean identity sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1 to rewrite cos⁡(2θ)\cos(2\theta) as 1−2sin⁡2(θ)1 - 2\sin^2(\theta) or 2cos⁡2(θ)−12\cos^2(\theta) - 1. Since we have a square root of 22 multiplying the sine function in the numerator, it might be beneficial to use the form 1−2sin⁡2(θ)1 - 2\sin^2(\theta) for cos⁡(2θ)\cos(2\theta) to potentially simplify the expression.
  3. Substitute Theta Value: Now, let's rewrite the limit using the chosen form of the double angle formula: lim⁡θ→−π4(1+2sin⁡(θ)1−2sin⁡2(θ))\lim_{\theta \rightarrow -\frac{\pi}{4}}\left(\frac{1+\sqrt{2}\sin(\theta)}{1 - 2\sin^2(\theta)}\right).
  4. Perform Operations: Next, we substitute θ=−π4\theta = -\frac{\pi}{4} into the sine function. We know that sin⁡(−π4)=−sin⁡(π4)=−12\sin\left(-\frac{\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}}. Let's substitute this value into the expression:\newlinelim⁡θ→−π41+2(−12)1−2(−12)2\lim_{\theta \rightarrow -\frac{\pi}{4}}\frac{1+\sqrt{2}\left(-\frac{1}{\sqrt{2}}\right)}{1 - 2\left(-\frac{1}{\sqrt{2}}\right)^2}.
  5. Simplify Further: Simplify the expression by performing the operations inside the limit: lim⁡θ→−π4(1−1)/(1−2(12))\lim_{\theta \rightarrow -\frac{\pi}{4}}(1 - 1)/(1 - 2(\frac{1}{2})).
  6. Identify Indeterminate Form: Further simplification gives us: lim⁡θ→−π401−1\lim_{\theta \rightarrow -\frac{\pi}{4}}\frac{0}{1 - 1}.
  7. Confirm Limit: We now have 00 in the numerator and 00 in the denominator, which is an indeterminate form. This means we cannot directly evaluate the limit by substitution. We need to find another way to evaluate the limit, possibly by factoring or using L'Hôpital's rule. However, since the numerator is already 00, the limit of the function as θ\theta approaches −π4-\frac{\pi}{4} is 00, provided that the denominator is not also 00 after simplification.
  8. Confirm Limit: We now have 00 in the numerator and 00 in the denominator, which is an indeterminate form. This means we cannot directly evaluate the limit by substitution. We need to find another way to evaluate the limit, possibly by factoring or using L'Hôpital's rule. However, since the numerator is already 00, the limit of the function as theta approaches −π4-\frac{\pi}{4} is 00, provided that the denominator is not also 00 after simplification.Looking back at the denominator, we see that after simplification, it is indeed 00, which confirms the indeterminate form 0/00/0. However, since the numerator is 00 regardless of the value of theta, and the denominator is not always 00 for values of theta near −π4-\frac{\pi}{4}, the limit of the function as theta approaches −π4-\frac{\pi}{4} is 00.

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