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 $\underset{x\to 3}{lim}\frac{x^3-9x}{x-3}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 0}{lim}\frac{x^3-9x}{x-3}\\ & =\underset{x\to 0}{lim}\frac{x(x^2-9)}{x-3}\\ & =\underset{x\to 0}{lim}\frac{x(x+3)(x-3)}{x-3}\\ & =\underset{x\to 0}{lim}x(x+3)\\ & =3(3+3)\\ & =18\end{array}$

Contoh soal 2

Nilai dari $\underset{x\to 4}{lim}\frac{\sqrt{x}-2}{x-4}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 4}{lim}\frac{\sqrt{x}-2}{x-4}\\ & =\underset{x\to 4}{lim}\frac{\sqrt{x}-2}{(\sqrt{x}{)}^2-2^2}\\ & =\underset{x\to 4}{lim}\frac{\sqrt{x}-2}{(\sqrt{x}-2)(\sqrt{x}+2)}\\ & =\underset{x\to 4}{lim}\frac{1}{\sqrt{x}+2}\\ & =\frac{1}{\sqrt{4}+2}\\ & =\frac{1}{4}\end{array}$

Contoh soal 3

Nilai dari $\underset{x\to 0}{lim}\frac{x}{\sqrt{4+x}-\sqrt{4-x}}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 0}{lim}\frac{x}{\sqrt{4+x}-\sqrt{4-x}}\\ & =\underset{x\to 0}{lim}\frac{x}{\sqrt{4+x}-\sqrt{4-x}}\times \frac{\sqrt{4+x}+\sqrt{4-x}}{\sqrt{4+x}+\sqrt{4-x}}\\ & =\underset{x\to 0}{lim}\frac{x(\sqrt{4+x}+\sqrt{4-x})}{2x}\\ & =\underset{x\to 0}{lim}\frac{\sqrt{4+x}+\sqrt{4-x}}{2}\\ & =\frac{\sqrt{4+0}+\sqrt{4-0}}{2}\\ & =\frac{2+2}{2}\\ & =2\end{array}$

Contoh soal 4

Nilai dari $\underset{x\to \sqrt{3}}{lim}\frac{x^2-3}{x-\sqrt{3}}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to \sqrt{3}}{lim}\frac{x^2-3}{x-\sqrt{3}}\\ & =\underset{x\to \sqrt{3}}{lim}\frac{(x-\sqrt{3})(x+\sqrt{3})}{x-\sqrt{3}}\\ & =\underset{x\to \sqrt{3}}{lim}x+\sqrt{3}\\ & =\sqrt{3}+\sqrt{3}\\ & =2\sqrt{3}\end{array}$

Contoh soal 5

Jika $\underset{x\to 0}{lim}\frac{g(x)}{x}=\frac{1}{2}$, maka nilai $\underset{x\to 0}{lim}\frac{g(x)}{\sqrt{1-x}-1}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 0}{lim}\frac{g(x)}{\sqrt{1-x}-1}\times \frac{\sqrt{1-x}+1}{\sqrt{1-x}+1}\\ & =\underset{x\to 0}{lim}\frac{g(x)(\sqrt{1-x}+1)}{-x}\\ & =\underset{x\to 0}{lim}\frac{g(x)}{-x}\times (\sqrt{1-x}+1)\\ & =-\frac{1}{2}(2)\\ & =-1\end{array}$

Contoh soal 6

Nilai dari $\underset{x\to 1}{lim}\frac{1-x}{2-\sqrt{x+3}}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 1}{lim}\frac{1-x}{2-\sqrt{x+3}}\times \frac{2+\sqrt{x+3}}{2+\sqrt{x+3}}\\ & =\underset{x\to 1}{lim}\frac{(1-x)(2+\sqrt{x+3})}{1-x}\\ & =\underset{x\to 1}{lim}2+\sqrt{x+3}\\ & =2+\sqrt{1+3}\\ & =2+2\\ & =4\end{array}$

Contoh soal 7

Nilai dari $\underset{x\to 3}{lim}\frac{x^2-9}{x^2+1}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 3}{lim}\frac{x^2-9}{x^2+1}\\ & =\frac{3^2-9}{3^2+1}\\ & =\frac{0}{10}\\ & =0\end{array}$

Contoh soal 8

Nilai dari $\underset{x\to 3}{lim}\frac{x^2-x-6}{x-3}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 3}{lim}\frac{x^2-x-6}{x-3}\\ & =\underset{x\to 3}{lim}\frac{(x-3)(x+2)}{x-3}\\ & =\underset{x\to 3}{lim}x+2\\ & =3+2\\ & =5\end{array}$

Contoh soal 9

Nilai dari $\underset{x\to 5}{lim}\frac{\sqrt{6x-5}-\sqrt{4x+5}}{x-5}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 5}{lim}\frac{\sqrt{6x-5}-\sqrt{4x+5}}{x-5}\\ & =\underset{x\to 5}{lim}\frac{\sqrt{6x-5}-\sqrt{4x+5}}{x-5}\times \frac{\sqrt{6x-5}+\sqrt{4x+5}}{\sqrt{6x-5}+\sqrt{4x+5}}\\ & =\underset{x\to 5}{lim}\frac{6x-5-(4x+5)}{x-5(\sqrt{6x-5}+\sqrt{4x+5})}\\ & =\underset{x\to 5}{lim}\frac{2x-10}{x-5(\sqrt{6x-5}+\sqrt{4x+5})}\\ & =\underset{x\to 5}{lim}\frac{2(x-5)}{x-5(\sqrt{6x-5}+\sqrt{4x+5})}\\ & =\underset{x\to 5}{lim}\frac{2}{(\sqrt{6x-5}+\sqrt{4x+5})}\\ & =\frac{2}{5+5}\\ & =\frac{2}{10}\\ & =\frac{1}{5}\end{array}$

Contoh soal 10

Nilai dari $\underset{x\to 1}{lim}\frac{x^3+2x^2-x-2}{x^2-5x+4}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 1}{lim}\frac{x^3+2x^2-x-2}{x^2-5x+4}\\ & =\underset{x\to 1}{lim}\frac{(x-1)(x^2+3x+2)}{(x-4)(x-1)}\\ & =\underset{x\to 1}{lim}\frac{x^2+3x+2}{x-4}\\ & =\frac{1^2+3.1+2}{1-4}\\ & =\frac{6}{-3}\\ & =-2\end{array}$

Demikianlah beberapa contoh soal dan pembahasan pada materi limit fungsi aljabar. Semoga bermanfaat.