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

${\displaystyle \underset{x\to \mathrm{\infty }}{lim}{\displaystyle \frac{a_0{x}^m+a_1{x}^{m-1}+a_2{x}^{m-2}+\cdots }{b_0{x}^n+b_1{x}^{n-1}+b_2{x}^{m-2}+\cdots }}=\{\begin{array}{ll}\mathrm{\infty }& \text{jika }m>n\\ {\displaystyle \frac{a_0}{b_0}}& \text{jika }m=n\\ 0& \text{jika }m<n\end{array}}$

---

Contoh Soal

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

$\begin{array}{rl}{\displaystyle \underset{x\to \mathrm{\infty }}{lim}{\displaystyle \frac{5x^4-3x^3+10x^2-20}{6x^4+7x^3-2x^2-3}}}& ={\displaystyle \underset{x\to \mathrm{\infty }}{lim}{\displaystyle \frac{{\displaystyle \frac{5x^4}{x^4}}-{\displaystyle \frac{3x^3}{x^4}}+{\displaystyle \frac{10x^2}{x^4}}-{\displaystyle \frac{20}{x^4}}}{{\displaystyle \frac{6x^4}{x^4}}+{\displaystyle \frac{7x^3}{x^4}}-{\displaystyle \frac{2x^2}{x^4}}-{\displaystyle \frac{3}{x^4}}}}}\\ & ={\displaystyle \underset{x\to \mathrm{\infty }}{lim}{\displaystyle \frac{5-{\displaystyle \frac{3}{x}}+{\displaystyle \frac{10}{x^2}}-{\displaystyle \frac{20}{x^4}}}{6+{\displaystyle \frac{7}{x}}-{\displaystyle \frac{2}{x^2}}-{\displaystyle \frac{3}{x^4}}}}}\\ & ={\displaystyle \frac{5-0+0-0}{6+0-0-0}}\\ & =\boxed{{\displaystyle \boxed{{\displaystyle {\displaystyle \frac{5}{6}}}}}}\end{array}$

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

---

BENTUK AKAR

${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\sqrt{a{x}^2+bx+c}-\sqrt{p{x}^2+qx+r}=\{\begin{array}{ll}\mathrm{\infty }& \text{jika }a>p\\ {\displaystyle \frac{b-q}{2\sqrt{a}}}& \text{jika }a=p\\ -\mathrm{\infty }& \text{jika }a<p\end{array}}$

\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}}}