Contoh Soal 2 Limit Fungsi Aljabar

$\begin{array}{l}\\ 6.& \text{Diketahui bahwa}f(x)=x^2-2,\\ & \text{maka nilai}\underset{h\to 0}{\text{Lim}}{\displaystyle \frac{f(x+h)-f(x)}{h}=....}\\ & \begin{array}{l}\\ \text{a}.{\displaystyle x^2-2}& & \text{d}.{\displaystyle x}\\ \\ \text{b}.{\displaystyle x^2}& \text{c}.{\displaystyle 2x}& \text{e}.{\displaystyle 2x-2}\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}\text{Diketahui}& \text{bahwa}f(x)=x^2-2,\\ \text{maka nila}& \text{i untuk}\\ \underset{h\to 0}{\text{Lim}}& {\displaystyle \frac{f(x+h)-f(x)}{h}}\\ & =\underset{h\to 0}{\text{Lim}}{\displaystyle \frac{((x+h{)}^2-2)-(x^2-2)}{h}}\\ & =\underset{h\to 0}{\text{Lim}}{\displaystyle {\displaystyle \frac{x^2+2xh+h^2-2-x^2+2}{h}}}\\ & =\underset{h\to 0}{\text{Lim}}{\displaystyle {\displaystyle \frac{2xh+h^2}{h}}}\\ & =\underset{h\to 0}{\text{Lim}}{\displaystyle (2x+h)}\\ & =2x\end{array}\end{array}$

$\begin{array}{l}\\ 7.& \text{Diketahui}f(x)=\sqrt{x-1},\\ & \text{maka nilai}\underset{h\to 0}{\text{Lim}}{\displaystyle \frac{f(2+h)-f(2)}{h}=....}\\ & \begin{array}{l}\\ \text{a}.{\displaystyle \frac{1}{2}}& & \text{d}.{\displaystyle 1}\\ \\ \text{b}.-{\displaystyle \frac{1}{2}}& \text{c}.{\displaystyle 0}& \text{e}.{\displaystyle -1}\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}\text{Diketa}& \text{hui bahwa}f(x)=\sqrt{x-1},\\ \text{maka}& \text{nilai untuk}\\ \underset{h\to 0}{\text{Lim}}& {\displaystyle \frac{f(2+h)-f(2)}{h}}\\ & =\underset{h\to 0}{\text{Lim}}{\displaystyle \frac{\sqrt{(2+h)-1}-\sqrt{2-1}}{h}}\\ & =\underset{h\to 0}{\text{Lim}}{\displaystyle {\displaystyle \frac{\sqrt{h+1}-1}{h}}}\\ & =\underset{h\to 0}{\text{Lim}}{\displaystyle {\displaystyle \frac{\sqrt{h+1}-1}{h}\times {\displaystyle \frac{\sqrt{h+1}+1}{\sqrt{h+1}+1}}}}\\ & =\underset{h\to 0}{\text{Lim}}{\displaystyle \frac{(h+1)-1}{h\times (\sqrt{h+1}+1)}}\\ & =\underset{h\to 0}{\text{Lim}}{\displaystyle \frac{1}{(\sqrt{h+1}+1)}}\\ & ={\displaystyle \frac{1}{\sqrt{0+1}+1}}\\ & ={\displaystyle \frac{1}{2}}\end{array}\end{array}$

$\begin{array}{l}\\ 8.& \mathbf{\text{(Mat Das SIMAK UI 2013)}}\\ & \text{Nilai}\underset{x\to 5}{\text{Lim}}{\displaystyle \frac{\sqrt{x+2\sqrt{x+1}}}{\sqrt{x-2\sqrt{x+1}}}=....}\\ & \begin{array}{l}\\ \text{a}.\sqrt{3}+\sqrt{2}& & \text{d}.5\\ \text{b}.5-2\sqrt{6}& \text{c}.2\sqrt{6}& \text{e}.5+2\sqrt{6}\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}\underset{x\to 5}{\text{Lim}}& {\displaystyle \frac{\sqrt{x+2\sqrt{x+1}}}{\sqrt{x-2\sqrt{x+1}}}}\\ & ={\displaystyle \frac{\sqrt{5+2\sqrt{5+1}}}{\sqrt{5-2\sqrt{5+1}}}}\\ & ={\displaystyle \frac{\sqrt{5+2\sqrt{6}}}{\sqrt{5-2\sqrt{6}}}}\\ & ={\displaystyle \frac{\sqrt{3+2+2\sqrt{3.2}}}{\sqrt{3+2-2\sqrt{3.2}}}}\\ & ={\displaystyle \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}}\\ & ={\displaystyle \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}\times {\displaystyle \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}}}}\\ & ={\displaystyle \frac{3+2+2\sqrt{6}}{3-2}}\\ & =5+2\sqrt{6}\end{array}\end{array}$

$\begin{array}{l}\\ 9.& \mathbf{\text{(Mat IPA SBMPTN 2014)}}\\ \\ & \text{Jika}\underset{x\to a}{\text{Lim}}{\displaystyle (f(x)+\frac{1}{g(x)})=4}\\ & \text{dan}\underset{x\to a}{\text{Lim}}{\displaystyle (f(x)-\frac{1}{g(x)})=-3,}\\ \\ & \text{maka nilai}\underset{x\to a}{\text{Lim}}{\displaystyle f(x).g(x)=....}\\ & \begin{array}{l}\\ \text{a}.{\displaystyle \frac{1}{14}}& & \text{d}.{\displaystyle \frac{4}{14}}\\ \\ \text{b}.{\displaystyle \frac{2}{14}}& \text{c}.{\displaystyle \frac{3}{14}}& \text{e}.{\displaystyle \frac{5}{14}}\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}& \text{Perhatikan bahwa},\\ & \begin{array}{l}\\ \underset{x\to a}{\text{Lim}}{\displaystyle (f(x)+\frac{1}{g(x)})=4}& & \\ & & \\ \underset{x\to a}{\text{Lim}}{\displaystyle (f(x)-\frac{1}{g(x)})=-3}& +& \\ & & \\ & & \\ 2\underset{x\to a}{\text{Lim}}{\displaystyle f(x)=1}& & \\ & & \\ \underset{x\to a}{\text{Lim}}f(x)={\displaystyle \frac{1}{2},}& & \end{array}\\ & \text{sehingga}\underset{x\to a}{\text{Lim}}f(g)={\displaystyle \frac{2}{7}}\\ & & \\ & \text{maka},\\ & \underset{x\to a}{\text{Lim}}f(x).g(x)={\displaystyle \frac{1}{2}\times \frac{2}{7}={\displaystyle \frac{2}{14}}}\end{array}\end{array}$