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f(x)=(2-cos^(2)(x))/(sin(2x)) We want to find lim_(x rarr(pi)/(2))f(x). What happens when we use direct substitution? Choose 1 answer: (A) The limit exists, and we found it! (B) The limit doesn't exist (probably an asymptote). (c) The result is indeterminate.

f(x)=2−cos⁡2(x)sin⁡(2x) f(x)=\frac{2-\cos ^{2}(x)}{\sin (2 x)} \newlineWe want to find lim⁡x→π2f(x) \lim _{x \rightarrow \frac{\pi}{2}} f(x) .\newlineWhat happens when we use direct substitution?\newlineChoose 11 answer:\newline(A) The limit exists, and we found it!\newline(B) The limit doesn't exist (probably an asymptote).\newline(C) The result is indeterminate.

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Full solution

Q. f(x)=2−cos⁡2(x)sin⁡(2x) f(x)=\frac{2-\cos ^{2}(x)}{\sin (2 x)} \newlineWe want to find lim⁡x→π2f(x) \lim _{x \rightarrow \frac{\pi}{2}} f(x) .\newlineWhat happens when we use direct substitution?\newlineChoose 11 answer:\newline(A) The limit exists, and we found it!\newline(B) The limit doesn't exist (probably an asymptote).\newline(C) The result is indeterminate.
  1. Direct Substitution: Let's first try direct substitution of x=π2x = \frac{\pi}{2} into the function f(x)=(2−cos⁡2(x))sin⁡(2x)f(x) = \frac{(2 - \cos^2(x))}{\sin(2x)} to see what we get.f(π2)=(2−cos⁡2(π2))sin⁡(2⋅π2)f\left(\frac{\pi}{2}\right) = \frac{(2 - \cos^2\left(\frac{\pi}{2}\right))}{\sin(2 \cdot \frac{\pi}{2})}
  2. Calculate Values: Now we calculate the cosine and sine values at x=π2x = \frac{\pi}{2}.cos⁡(π2)=0\cos(\frac{\pi}{2}) = 0, so cos⁡2(π2)=02=0\cos^2(\frac{\pi}{2}) = 0^2 = 0.sin⁡(π2)=1\sin(\frac{\pi}{2}) = 1, so sin⁡(2⋅π2)=sin⁡(π)=0\sin(2 \cdot \frac{\pi}{2}) = \sin(\pi) = 0.
  3. Substitute Values: Substitute these values into the function.\newlinef(π2)=(2−0)/0f(\frac{\pi}{2}) = (2 - 0) / 0\newlinef(π2)=20f(\frac{\pi}{2}) = \frac{2}{0}\newlineThis results in a division by zero, which is undefined.
  4. Division by Zero: Since we have a division by zero, the limit does not exist through direct substitution, and we encounter an indeterminate form of the type 20\frac{2}{0}, which suggests a possible asymptote or that the limit does not exist.

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