Limit Tak Hingga Fungsi Aljabar

Limit tak Hingga Fungsi Aljabar

Bentuk 1: \(\displaystyle \lim_{ x\to \infty }\sqrt{ax^2+bx+c}-\sqrt{px^2+qx+r}=...\)

Penyelesaian:

\(\begin{array}{rcl}&=&\displaystyle \lim_{ x\to \infty}\sqrt{ax^2+bx+c}-\sqrt{px^2+qx+r}\times \left (\frac{\sqrt{ax^2+bx+c}+\sqrt{px^2+qx+r}}{\sqrt{ax^2+bx+c}+\sqrt{px^2+qx+r}}\right )\\&=&\displaystyle \lim_{x \to \infty }\frac{\left ( ax^2+bx+c \right )-\left ( px^2+qx+r \right )}{\sqrt{ax^2+bx+c}-\sqrt{px^2+qx+r}}\\&=&\displaystyle \lim_{x \to \infty }\frac{(a-p)x^2+(b-q)x+(c-r)}{\sqrt{ax^2+bx+c}+\sqrt{px^2+qx+r}} \end{array}\)

Bila $a = p$

\(\begin{array}{rcl}&=&\displaystyle \lim_{x \to \infty }\frac{(a-a)x^2+(b-q)x+(c-r)}{\sqrt{ax^2+bx+c}+\sqrt{ax^2+qx+r}} \\&=&\displaystyle \lim_{x \to \infty }\frac{(b-q)x+(c-r)}{\sqrt{ax^2+bx+c}+\sqrt{ax^2+qx+r}}\\&=&\displaystyle \lim_{x \to \infty }\frac{(b-q)}{2\sqrt{a}}\end{array}\)

Bila $a > p$

\(\begin{array}{rcl}&=&\displaystyle \lim_{x \to \infty }\frac{(a-p)x^2+(b-q)x+(c-r)}{\sqrt{ax^2+bx+c}+\sqrt{px^2+qx+r}} \\&=&\displaystyle \lim_{x \to \infty }\frac{kx^2+(b-q)x+(c-r)}{\sqrt{ax^2+bx+c}+\sqrt{px^2+qx+r}}\\&=&\infty\end{array}\)

Bila $a < p$

\(\begin{array}{rcl}&=&\displaystyle \lim_{x \to \infty }\frac{(a-p)x^2+(b-q)x+(c-r)}{\sqrt{ax^2+bx+c}+\sqrt{px^2+qx+r}} \\&=&\displaystyle \lim_{x \to \infty }\frac{-kx^2+(b-q)x+(c-r)}{\sqrt{ax^2+bx+c}+\sqrt{px^2+qx+r}}\\&=&-\infty\end{array}\)

Bentuk 2:\(\displaystyle \lim_{ x\to \infty }\sqrt[3]{ax^3+bx^2+cx+d}-\sqrt[3]{px^3+qx^2+rx+s}=...\)

Penyelesaian:

\(\begin{array}{rcl}&=&\displaystyle \lim_{ x\to \infty}\sqrt[3]{ax^3+bx^2+cx+d}-\sqrt[3]{px^3+qx^2+rx+s}\times \left ( \frac{\sqrt[3]{\left ( ax^3+bx^2+cx+d \right )^{2}}+\sqrt[3]{\left ( ax^3+bx^2+cx+d \right ){\left ( px^3+qx^2+rx+s \right )}}+\sqrt[3]{\left ( px^3+qx^2+rx+s \right )^{2}}}{\sqrt[3]{\left ( ax^3+bx^2+cx+d \right )^{2}}+\sqrt[3]{\left ( ax^3+bx^2+cx+d \right )\left ( px^3+qx^2+rx+s \right )}+\sqrt[3]{\left ( px^3+qx^2+rx+s \right )^{2}}} \right )\\&=&\displaystyle \lim_{ x\to \infty}\frac{\left ( ax^3+bx^2+cx+d \right )-\left ( px^3+qx^2+rx+s \right )}{\sqrt[3]{(ax^3+bx^2+cx+d)^{2}}+\sqrt[3]{(ax^3+bx^2+cx+d)(px^3+qx^2+rx+s)}+\sqrt[3]{(px^3+qx^2+rx+s)^{2}}}\\&=&\displaystyle \lim_{ x\to \infty}\frac{(a-p)x^3+(b-q)x^2+(c-r)x+(d-s) }{\sqrt[3]{(ax^3+bx^2+cx+d)^{2}}+\sqrt[3]{(ax^3+bx^2+cx+d)(px^3+qx^2+rx+s)}+\sqrt[3]{(px^3+qx^2+rx+s)^{2}}} \end{array}\)

Bila $a = p$

\(\begin{array}{rcl}&=&\displaystyle \lim_{ x\to \infty}\frac{(a-a)x^3+(b-q)x^2+(c-r)x+(d-s) }{\sqrt[3]{(ax^3+bx^2+cx+d)^{2}}+\sqrt[3]{(ax^3+bx^2+cx+d)(ax^3+qx^2+rx+s)}+\sqrt[3]{(ax^3+qx^2+rx+s)^{2}}}\\&=&\displaystyle \lim_{ x\to \infty}\frac{(b-q)x^2+(c-r)x+(d-s) }{\sqrt[3]{(ax^3)^{2}}+\sqrt[3]{(ax^3)(ax^3)}+\sqrt[3]{(ax^3)^{2}}}\\&=&\displaystyle \lim_{ x\to \infty}\frac{(b-q)x^2+(c-r)x+(d-s) }{\sqrt[3]{(a^2x^6)}+\sqrt[3]{(a^2x^6)}+\sqrt[3]{(a^2x^6)}}\\&=&\displaystyle \lim_{ x\to \infty}\frac{(b-q)x^2+(c-r)x+(d-s) }{\sqrt[3]{(a)^{2}}x^2+\sqrt[3]{(a)^{2}}x^2+\sqrt[3]{(a)^{2}}x^2} \\&=&\displaystyle \lim_{ x\to \infty}\frac{(b-q)x^2+(c-r)x+(d-s) }{3\sqrt[3]{a^{2}}x^2}\\&=&\frac{b-q }{3\sqrt[3]{a^{2}}} \end{array}\)

Bila $a > p$

\(\begin{array}{rcl}&=&\displaystyle \lim_{ x\to \infty}\frac{(a-p)x^3+(b-q)x^2+(c-r)x+(d-s) }{\sqrt[3]{(ax^3+bx^2+cx+d)^{2}}+\sqrt[3]{(ax^3+bx^2+cx+d)(px^3+qx^2+rx+s)}+\sqrt[3]{(px^3+qx^2+rx+s)^{2}}}\\&=&\displaystyle \lim_{ x\to \infty}\frac{kx^3+(b-q)x^2+(c-r)x+(d-s) }{\sqrt[3]{(ax^3+bx^2+cx+d)^{2}}+\sqrt[3]{(ax^3+bx^2+cx+d)(ax^3+qx^2+rx+s)}+\sqrt[3]{(ax^3+qx^2+rx+s)^{2}}}\\&=&\displaystyle \lim_{ x\to \infty}\frac{kx^3+(b-q)x^2+(c-r)x+(d-s) }{lx^2}\\&=&\displaystyle \lim_{ x\to \infty}\frac{kx^3 }{lx^2}\\&=&\infty \end{array}\)

Bila $a < p$

\(\begin{array}{rcl}&=&\displaystyle \lim_{ x\to \infty}\frac{(a-p)x^3+(b-q)x^2+(c-r)x+(d-s) }{\sqrt[3]{(ax^3+bx^2+cx+d)^{2}}+\sqrt[3]{(ax^3+bx^2+cx+d)(px^3+qx^2+rx+s)}+\sqrt[3]{(px^3+qx^2+rx+s)^{2}}}\\&=&\displaystyle \lim_{ x\to \infty}\frac{-kx^3+(b-q)x^2+(c-r)x+(d-s) }{\sqrt[3]{(ax^3+bx^2+cx+d)^{2}}+\sqrt[3]{(ax^3+bx^2+cx+d)(ax^3+qx^2+rx+s)}+\sqrt[3]{(ax^3+qx^2+rx+s)^{2}}}\\&=&\displaystyle \lim_{ x\to \infty}\frac{-kx^3+(b-q)x^2+(c-r)x+(d-s) }{lx^2}\\&=&\displaystyle \lim_{ x\to \infty}\frac{-kx^3 }{lx^2}\\&=&-\infty \end{array}\)