Limit Tak Hingga

Limit tak hingga artinya bentuk limit dimana x menuju tak hingga. Limit tak hingga biasanya berbentuk pecahan atau bentuk akar.

BENTUK PECAHAN

\(\displaystyle\lim_{x\to\infty}\dfrac{a_0x^m+a_1x^{m-1}+a_2x^{m-2}+\cdots}{b_0x^n+b_1x^{n-1}+b_2x^{m-2}+\cdots}=\begin{cases} \infty&\text{jika }m\gt n\\[8pt] \dfrac{a_0}{b_0}&\text{jika }m=n\\[8pt] 0&\text{jika }m\lt n \end{cases}\)

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Contoh Soal

\displaystyle\lim_{x\to\infty}\dfrac{5x^4-3x^3+10x^2-20}{6x^4+7x^3-2x^2-3}= ....

\(\begin{aligned} \displaystyle\lim_{x\to\infty}\dfrac{5x^4-3x^3+10x^2-20}{6x^4+7x^3-2x^2-3}&=\displaystyle\lim_{x\to\infty}\dfrac{\dfrac{5x^4}{x^4}-\dfrac{3x^3}{x^4}+\dfrac{10x^2}{x^4}-\dfrac{20}{x^4}}{\dfrac{6x^4}{x^4}+\dfrac{7x^3}{x^4}-\dfrac{2x^2}{x^4}-\dfrac3{x^4}}\\[22pt] &=\displaystyle\lim_{x\to\infty}\dfrac{5-\dfrac3x+\dfrac{10}{x^2}-\dfrac{20}{x^4}}{6+\dfrac7x-\dfrac2{x^2}-\dfrac3{x^4}}\\[22pt] &=\dfrac{5-0+0-0}{6+0-0-0}\\ &=\boxed{\boxed{\dfrac56}} \end{aligned}\)

Jadi, \displaystyle\lim_{x\to\infty}\dfrac{5x^4-3x^3+10x^2-20}{6x^4+7x^3-2x^2-3}=\dfrac56.

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BENTUK AKAR

\(\displaystyle\lim_{x\to\infty}\sqrt{ax^2+bx+c}-\sqrt{px^2+qx+r}=\begin{cases} \infty&\text{jika }a\gt p\\[8pt] \dfrac{b-q}{2\sqrt{a}}&\text{jika }a=p\\[8pt] -\infty&\text{jika }a\lt p \end{cases}\)

\displaystyle\lim_{x\to\infty}\sqrt[n]{ax^n+bx^{n-1}+\cdots}-\sqrt[n]{ax^n+px^{n-1}+\cdots}=\dfrac{b-p}{2\sqrt[n]{a^{n-1}}}