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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$$\begin{array}{rl}{\displaystyle \underset{x\to \mathrm{\infty }}{lim}{({\left({\displaystyle \frac{1}{2}}\right)}^{3x}+{\left({\displaystyle \frac{1}{2}}\right)}^x)}^{\frac{1}{x^2}}}& ={\displaystyle \underset{x\to \mathrm{\infty }}{lim}{({\displaystyle \frac{1}{2^{3x}}}+{\displaystyle \frac{1}{2^x}})}^{\frac{1}{x^2}}}\\ & ={\displaystyle \underset{x\to \mathrm{\infty }}{lim}{\left({\displaystyle \frac{2^{2x}+1}{2^{3x}}}\right)}^{\frac{1}{x^2}}}\\ & ={\displaystyle \underset{x\to \mathrm{\infty }}{lim}{\left({\displaystyle \frac{2^{2x}}{2^{3x}}}\right)}^{\frac{1}{x^2}}}\\ & ={\displaystyle \underset{x\to \mathrm{\infty }}{lim}{\left(2^{-x}\right)}^{\frac{1}{x^2}}}\\ & ={\displaystyle \underset{x\to \mathrm{\infty }}{lim}{2}^{-\frac{1}{x}}}\\ & =2^{-0}\\ & =\boxed{{\displaystyle \boxed{{\displaystyle 1}}}}\end{array}$$

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