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g(x)=(cos(x)-sin(x))/(cos(2x)) We want to find lim_(x rarr(pi)/(4))g(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.

g(x)=cos⁡(x)−sin⁡(x)cos⁡(2x) g(x)=\frac{\cos (x)-\sin (x)}{\cos (2 x)} \newlineWe want to find lim⁡x→π4g(x) \lim _{x \rightarrow \frac{\pi}{4}} g(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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Q. g(x)=cos⁡(x)−sin⁡(x)cos⁡(2x) g(x)=\frac{\cos (x)-\sin (x)}{\cos (2 x)} \newlineWe want to find lim⁡x→π4g(x) \lim _{x \rightarrow \frac{\pi}{4}} g(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: First, let's try direct substitution of x=π4x = \frac{\pi}{4} into the function g(x)g(x) to see what happens.\newlineg(π4)=cos⁡(π4)−sin⁡(π4)cos⁡(2⋅π4)g\left(\frac{\pi}{4}\right) = \frac{\cos\left(\frac{\pi}{4}\right) - \sin\left(\frac{\pi}{4}\right)}{\cos(2 \cdot \frac{\pi}{4})}
  2. Calculate Trigonometric Values: Now, we calculate the values of cos⁡(π4)\cos(\frac{\pi}{4}), sin⁡(π4)\sin(\frac{\pi}{4}), and cos⁡(2×π4)\cos(2 \times \frac{\pi}{4}).\newlinecos⁡(π4)=22\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\newlinesin⁡(π4)=22\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\newlinecos⁡(2×π4)=cos⁡(π2)=0\cos(2 \times \frac{\pi}{4}) = \cos(\frac{\pi}{2}) = 0
  3. Substitute Values into Function: Substitute these values into the function g(x)g(x).g(π4)=(22−22)/0g(\frac{\pi}{4}) = (\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}) / 0This simplifies to:g(π4)=00g(\frac{\pi}{4}) = \frac{0}{0}
  4. Indeterminate Form: The result of the substitution is 0/00/0, which is an indeterminate form. This means that direct substitution does not give us the limit, and we need to apply other techniques to find the limit.

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