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Find lim_(h rarr0)(4sin((pi)/(2)+h)-4sin((pi)/(2)))/(h). Choose 1 answer: (A) 0 (B) 1 (C) 4 D The limit doesn't exist

Find lim⁡h→04sin⁡(π2+h)−4sin⁡(π2)h \lim _{h \rightarrow 0} \frac{4 \sin \left(\frac{\pi}{2}+h\right)-4 \sin \left(\frac{\pi}{2}\right)}{h} .\newlineChoose 11 answer:\newline(A) 00\newline(B) 11\newline(C) 44\newline(D) The limit doesn't exist

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Full solution

Q. Find lim⁡h→04sin⁡(π2+h)−4sin⁡(π2)h \lim _{h \rightarrow 0} \frac{4 \sin \left(\frac{\pi}{2}+h\right)-4 \sin \left(\frac{\pi}{2}\right)}{h} .\newlineChoose 11 answer:\newline(A) 00\newline(B) 11\newline(C) 44\newline(D) The limit doesn't exist
  1. Identify Limit: Identify the limit to solve: lim⁡h→04sin⁡(π2+h)−4sin⁡(π2)h\lim_{h \to 0}\frac{4\sin\left(\frac{\pi}{2}+h\right)-4\sin\left(\frac{\pi}{2}\right)}{h}.
  2. Recognize Sin Value: Recognize that sin⁡(π2)=1\sin\left(\frac{\pi}{2}\right) = 1, so 4sin⁡(π2)=44\sin\left(\frac{\pi}{2}\right) = 4.
  3. Substitute Sin Value: Substitute sin⁡(π2)\sin\left(\frac{\pi}{2}\right) with 11 in the expression: lim⁡h→0(4sin⁡(π2+h)−4h)\lim_{h \to 0}\left(\frac{4\sin\left(\frac{\pi}{2}+h\right)-4}{h}\right).
  4. Apply Limit to Sine: Apply the limit to the sine function directly: lim⁡h→04(sin⁡(π2+h)−1)h.\lim_{h \to 0}\frac{4(\sin(\frac{\pi}{2}+h)-1)}{h}.
  5. Use Approximation: Use the fact that sin⁡(π2+h)\sin\left(\frac{\pi}{2}+h\right) is approximately 11 when hh is close to 00: lim⁡h→04(1−1)/h\lim_{h \rightarrow 0}4\left(1-1\right)/h.

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