Exercise Zone : Limit [4]

Table of Contents

Berikut ini adalah kumpulan soal mengenai Limit. Jika ingin bertanya soal, silahkan gabung ke grup Telegram, Signal, Discord, atau WhatsApp.

Tipe:

StandarSNBTHOTS

No.

Tentukan nilai limit dengan "memfaktorkan / bentuk akar sekawan" berikut :

1. \displaystyle\lim_{x\to2}\dfrac{x^2+2x-8}{x-2} = ....
2. \displaystyle\lim_{x\to2}\dfrac{\sqrt{x}(x-2)}{\sqrt{x}-\sqrt2} = ....

Latihan Pra UKK Semester Genap Kelas XI

ALTERNATIF PENYELESAIAN

1. 
   \begin{aligned} \lim_{x\to2}\frac{x^2+2x-8}{x-2}&=\lim_{x\to2}\frac{(x+4)(x-2)}{x-2}\\[3.7pt] &=\lim_{x\to2}(x+4)\\ &=2+4\\ &=\boxed{\boxed{6}} \end{aligned}
2. 
   \begin{aligned} \lim_{x\to2}\frac{\sqrt{x}(x-2)}{\sqrt{x}-\sqrt2}&=\lim_{x\to2}\frac{\sqrt{x}(x-2)}{\sqrt{x}-\sqrt2}\cdot\frac{\sqrt{x}+\sqrt2}{\sqrt{x}+\sqrt2}\\[3.7pt] &=\lim_{x\to2}\frac{\sqrt{x}(x-2)\left(\sqrt{x}+\sqrt2\right)}{x-2}\\[3.7pt] &=\lim_{x\to2}\sqrt{x}\left(\sqrt{x}+\sqrt2\right)\\ &=\sqrt{2}\left(\sqrt{2}+\sqrt2\right)\\ &=\sqrt{2}\left(2\sqrt2\right)\\ &=2\cdot2\\ &=\boxed{\boxed{4}} \end{aligned}

Jadi,

1. \displaystyle\lim_{x\to2}\dfrac{x^2+2x-8}{x-2}=6
2. \displaystyle\lim_{x\to2}\dfrac{\sqrt{x}(x-2)}{\sqrt{x}-\sqrt2}=4

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No.

\displaystyle\lim_{x\to4}\dfrac{3x^2-48x}{x^2-16}= ....

latiseducation.com

ALTERNATIF PENYELESAIAN

\begin{aligned} \lim_{x\to4}\dfrac{3x^2-48x}{x^2-16}&=\lim_{x\to4}\dfrac{3x\color{red}{\left(x^2-16\right)}}{\color{red}{x^2-16}}\\ &=\lim_{x\to4}3x\\ &=3(4)\\ &=\boxed{\boxed{12}} \end{aligned}

Jadi, \displaystyle\lim_{x\to4}\dfrac{3x^2-48x}{x^2-16}=12.

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No.

Nilai dari \displaystyle\lim_{x\to4}\sqrt[3]{3x^2+7x-12}+\displaystyle\lim_{x\to5}\left(\sqrt{3x^2-11}-3x\right) adalah ....

latiseducation.com

ALTERNATIF PENYELESAIAN

Menggunakan teknik substitusi langsung. \begin{aligned} \displaystyle\lim_{x\to4}\sqrt[3]{3x^2+7x-12}+\displaystyle\lim_{x\to5}\left(\sqrt{3x^2-11}-3x\right)&=\sqrt[3]{3(4)^2+7(4)-12}+\left(\sqrt{3(5)^2-11}-3(5)\right)\\ &=\sqrt[3]{48+28-12}+\sqrt{75-11}-15\\ &=\sqrt[3]{64}+\sqrt{64}-15\\ &=4+8-15\\ &=\boxed{\boxed{-3}} \end{aligned}

Jadi, nilai dari \displaystyle\lim_{x\to4}\sqrt[3]{3x^2+7x-12}+\displaystyle\lim_{x\to5}\left(\sqrt{3x^2-11}-3x\right) adalah −3.

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No.

Diketahui
$f(x)=\begin{cases}3x-p,\ x\leq2\\2x+1,\ x\gt2\end{cases}$
Agar \displaystyle\lim_{x\to2}f(x) mempunyai nilai, maka p = ....

1. −2
2. −1
3. 0

4. 1
5. 2

UNBK SMA IPA 2018

ALTERNATIF PENYELESAIAN

Berdasarkan definisi limit, agar \displaystyle\lim_{x\to2}f(x) mempunyai nilai maka Limit kiri = Limit Kanan. Secara simbol dituliskan
\displaystyle\lim_{x\to2^+}f(x)=\displaystyle\lim_{x\to2^-}f(x)=L

Limit kiri: \begin{aligned}\displaystyle\lim_{x\to2^-}f(x)&=\displaystyle\lim_{x\to2^-}(3x-p)\\ &=3(2)-p\\ &=6-p \end{aligned} Limit kanan: \begin{aligned}\displaystyle\lim_{x\to2^+}f(x)&=\displaystyle\lim_{x\to2^+}(2x+1)\\ &=2(2)+1\\ &=5 \end{aligned} \begin{aligned} 6-p&=5\\ 6-5&=p\\ p&=\boxed{\boxed{1}} \end{aligned}

Jadi, p = 1.
JAWAB: D

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