HOTS Zone : Limit Tak Hingga

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Tipe:

Standar SBMPTN HOTS

No.

\displaystyle\lim_{x\to\infty}\left(\left(\dfrac12\right)^{3x}+\left(\dfrac12\right)^x\right)^{\frac1{x^2}}= ....

1. -4
2. -2
   
3. 1
   

4. 2
5. 3
   

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\begin{aligned} \displaystyle\lim_{x\to\infty}\left(\left(\dfrac12\right)^{3x}+\left(\dfrac12\right)^x\right)^{\frac1{x^2}}&=\displaystyle\lim_{x\to\infty}\left(\dfrac1{2^{3x}}+\dfrac1{2^x}\right)^{\frac1{x^2}}\\[8pt] &=\displaystyle\lim_{x\to\infty}\left(\dfrac{2^{2x}+1}{2^{3x}}\right)^{\frac1{x^2}}\\[8pt] &=\displaystyle\lim_{x\to\infty}\left(\dfrac{2^{2x}}{2^{3x}}\right)^{\frac1{x^2}}\\[8pt] &=\displaystyle\lim_{x\to\infty}\left(2^{-x}\right)^{\frac1{x^2}}\\[8pt] &=\displaystyle\lim_{x\to\infty}2^{-\frac1x}\\[8pt] &=2^{-0}\\[6pt] &=\boxed{\boxed{1}} \end{aligned}

Jadi, \displaystyle\lim_{x\to\infty}\left(\left(\dfrac12\right)^{3x}+\left(\dfrac12\right)^x\right)^{\frac1{x^2}}=1.