Pembahasan Contoh Soal Materi Limit Fungsi Aljabar
Berikut ini mimin sajikan beberapa contoh soal dan pembahasan pada materi limit fungsi aljabar. Selamat membaca, sobat. Semoga bermanfaat.
Contoh soal 1
Nilai dari $\lim\limits_{x \to 3} \frac{x^3-9x}{x-3}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to 0} \frac{x^3-9x}{x-3} \\&= \lim\limits_{x \to 0} \frac{x(x^2-9)}{x-3}\\&= \lim\limits_{x \to 0} \frac{x(x+3)(x-3)}{x-3} \\&= \lim\limits_{x \to 0} x(x+3) \\&= 3(3+3)\\&=18 \end{aligned}$
Contoh soal 2
Nilai dari $\lim\limits_{x \to 4} \frac{\sqrt{x}-2}{x-4}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to 4} \frac{\sqrt{x}-2}{x-4} \\&= \lim\limits_{x \to 4} \frac{\sqrt{x}-2}{(\sqrt{x})^2-2^2}\\&= \lim\limits_{x \to 4} \frac{\sqrt{x}-2}{(\sqrt{x}-2)(\sqrt{x}+2)} \\&= \lim\limits_{x \to 4} \frac{1}{\sqrt{x}+2} \\&= \frac{1}{\sqrt{4}+2}\\&=\frac{1}{4} \end{aligned}$
Contoh soal 3
Nilai dari $\lim\limits_{x \to 0} \frac{x}{\sqrt{4+x}-\sqrt{4-x}}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to 0} \frac{x}{\sqrt{4+x}-\sqrt{4-x}}\\&= \lim\limits_{x \to 0} \frac{x}{\sqrt{4+x}-\sqrt{4-x}} \times \frac{\sqrt{4+x}+\sqrt{4-x}}{\sqrt{4+x}+\sqrt{4-x}} \\&= \lim\limits_{x \to 0} \frac{x(\sqrt{4+x}+\sqrt{4-x})}{2x} \\&= \lim\limits_{x \to 0} \frac{\sqrt{4+x}+\sqrt{4-x}}{2}\\&= \frac{\sqrt{4+0}+\sqrt{4-0}}{2}\\&= \frac{2+2}{2}\\&=2 \end{aligned}$
Contoh soal 4
Nilai dari $\lim\limits_{x \to \sqrt{3}} \frac{x^2-3}{x-\sqrt{3}}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to \sqrt{3}} \frac{x^2-3}{x-\sqrt{3}}\\&= \lim\limits_{x \to \sqrt{3}} \frac{(x-\sqrt{3})(x+\sqrt{3})}{x-\sqrt{3}} \\&= \lim\limits_{x \to \sqrt{3}} x+\sqrt{3} \\&= \sqrt{3}+\sqrt{3}\\&= 2 \sqrt{3} \end{aligned}$
Contoh soal 5
Jika $\lim\limits_{x \to 0} \frac{g(x)}{x} =\frac{1}{2}$, maka nilai $\lim\limits_{x \to 0} \frac{g(x)}{\sqrt{1-x}-1} $ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to 0} \frac{g(x)}{\sqrt{1-x}-1} \times \frac{\sqrt{1-x}+1}{\sqrt{1-x}+1}\\&= \lim\limits_{x \to 0} \frac{g(x) (\sqrt{1-x}+1)}{-x} \\&= \lim\limits_{x \to 0} \frac{g(x)}{-x} \times (\sqrt{1-x}+1) \\&= -\frac{1}{2}(2)\\&=-1 \end{aligned}$
Contoh soal 6
Nilai dari $\lim\limits_{x \to 1} \frac{1-x}{2-\sqrt{x+3}}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to 1} \frac{1-x}{2-\sqrt{x+3}} \times \frac{2+\sqrt{x+3}}{2+\sqrt{x+3}}\\&= \lim\limits_{x \to 1} \frac{(1-x) (2+\sqrt{x+3})}{1-x}\\&= \lim\limits_{x \to 1} 2+\sqrt{x+3}\\&= 2+\sqrt{1+3}\\&= 2+2\\&= 4 \end{aligned}$
Contoh soal 7
Nilai dari $\lim\limits_{x \to 3} \frac{x^2-9}{x^2+1}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to 3} \frac{x^2-9}{x^2+1} \\&= \frac{3^2-9}{3^2+1}\\&=\frac{0}{10}\\&=0 \end{aligned}$
Contoh soal 8
Nilai dari $\lim\limits_{x \to 3} \frac{x^2-x-6}{x-3}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to 3} \frac{x^2-x-6}{x-3} \\&= \lim\limits_{x \to 3} \frac{(x-3)(x+2)}{x-3}\\&= \lim\limits_{x \to 3} x+2\\&= 3+2\\&=5 \end{aligned}$
Contoh soal 9
Nilai dari $\lim\limits_{x \to 5} \frac{\sqrt{6x-5}-\sqrt{4x+5}}{x-5}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to 5} \frac{\sqrt{6x-5}-\sqrt{4x+5}}{x-5}\\&= \lim\limits_{x \to 5} \frac{\sqrt{6x-5}-\sqrt{4x+5}}{x-5} \times \frac{\sqrt{6x-5}+\sqrt{4x+5}}{\sqrt{6x-5}+\sqrt{4x+5}}\\&= \lim\limits_{x \to 5} \frac{6x-5-(4x+5)}{x-5 (\sqrt{6x-5}+\sqrt{4x+5})} \\&= \lim\limits_{x \to 5} \frac{2x-10}{x-5 (\sqrt{6x-5}+\sqrt{4x+5})} \\&= \lim\limits_{x \to 5} \frac{2(x-5)}{x-5 (\sqrt{6x-5}+\sqrt{4x+5})}\\&= \lim\limits_{x \to 5} \frac{2}{(\sqrt{6x-5}+\sqrt{4x+5})}\\&= \frac{2}{5+5}\\&= \frac{2}{10}\\&=\frac{1}{5} \end{aligned}$
Contoh soal 10
Nilai dari $\lim\limits_{x \to 1} \frac{x^3+2x^2-x-2}{x^2-5x+4}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to 1} \frac{x^3+2x^2-x-2}{x^2-5x+4}\\&= \lim\limits_{x \to 1} \frac{(x-1)(x^2+3x+2)}{(x-4)(x-1)}\\&= \lim\limits_{x \to 1} \frac{x^2+3x+2}{x-4}\\&= \frac{1^2+3.1+2}{1-4}\\&= \frac{6}{-3}\\&=-2 \end{aligned}$
Demikianlah beberapa contoh soal dan pembahasan pada materi limit fungsi aljabar. Semoga bermanfaat.
