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prod_(k=1)^(2)k ×lim_(x rarr0)(e^(16 x)-1)/(x)+sum_(k=0)^(2)sin((2pi k)/(3))

∏k=12k×lim⁡x→0e16x−1x+∑k=02sin⁡(2πk3) \prod_{k=1}^{2} k \times \lim _{x \rightarrow 0} \frac{e^{16 x}-1}{x}+\sum_{k=0}^{2} \sin \left(\frac{2 \pi k}{3}\right)

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Q. ∏k=12k×lim⁡x→0e16x−1x+∑k=02sin⁡(2πk3) \prod_{k=1}^{2} k \times \lim _{x \rightarrow 0} \frac{e^{16 x}-1}{x}+\sum_{k=0}^{2} \sin \left(\frac{2 \pi k}{3}\right)
  1. Calculate Product: Calculate the product from k=1k=1 to 22 of kk.∏k=12k=1×2=2\prod_{k=1}^{2}k = 1 \times 2 = 2
  2. Evaluate Limit: Evaluate the limit as xx approaches 00 of e16x−1x\frac{e^{16x}-1}{x}.lim⁡x→0e16x−1x=16\lim_{x \to 0}\frac{e^{16 x}-1}{x} = 16, because the limit of eax−1x\frac{e^{ax}-1}{x} as xx approaches 00 is aa.
  3. Calculate Sum: Calculate the sum from k=0k=0 to 22 of sin⁡(2πk3)\sin\left(\frac{2\pi k}{3}\right).∑k=02sin⁡(2πk3)=sin⁡(0)+sin⁡(2π3)+sin⁡(4π3)\sum_{k=0}^{2}\sin\left(\frac{2\pi k}{3}\right) = \sin(0) + \sin\left(\frac{2\pi}{3}\right) + \sin\left(\frac{4\pi}{3}\right)=0+(32)+(−32)= 0 + \left(\frac{\sqrt{3}}{2}\right) + \left(-\frac{\sqrt{3}}{2}\right)=0= 0
  4. Add Results: Add the results from the previous steps.\newline2+16+0=182 + 16 + 0 = 18

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