ข้อสอบ PAT 1 - มีนาคม 2560

$$\begin{gathered} \mathrm{จากโจทย์} \underset{x\to 3}{\lim }\frac{3^x·x-3^{\left( x+1\right)}}{\sqrt[3]{x-2}-1} \mathrm{เท่ากับเท่าใด} \\ \mathrm{แทน} x=3 \underset{x\to 3}{\lim }\frac{3^x·x-3^{\left( x+1\right)}}{\sqrt[3]{x-2}-1} = \frac{3^3·3-3^{\left( 3+1\right)}}{\sqrt[3]{3-2}-1} \\ = \frac{3^4-3^4}{1-1} = \frac{0}{0} \end{gathered}$$

$$\begin{gathered} \mathrm{แสดงว่า} \mathrm{ต้องจัดรูปใหม่} \mathrm{เพื่อหาค่า} limit \\ \underset{x\to 3}{\lim }\frac{3^x·x-3^{\left(x+1\right)}}{\sqrt[3]{x-2}-1} = \underset{x\to 3}{\lim }\frac{3^x·x-3^x·3}{\sqrt[3]{x-2}-1} \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\underset{x\to 3}{\lim }\frac{3^x\left(x-3\right)}{\sqrt[3]{x-2}-1} \to \boxed{1} \end{gathered}$$

$$\begin{gathered} \mathrm{กำหนดให้} a= \sqrt[3]{x-2} \to a^3=x-2 \\ b= 1 \to b^3=1 \\ \mathrm{สูตร} a^3-b^3 = \left(a-b\right)\left(a^2+ab+b^2\right) \\ \mathrm{จาก} \boxed{1} \mathrm{จะได้} \underset{x\to 3}{\lim }\frac{3^x\left(x-3\right)}{\sqrt[3]{x-2}-1} = \underset{x\to 3}{\lim }\frac{3^x\left(x-3\right)}{a-b} \\ \mathrm{นำ} a^2+ab+b^2 \mathrm{คูณทั้งเศษและส่วน} \end{gathered}$$

$$\begin{gathered} \mathrm{จะได้} = \underset{x\to 3}{\lim }\frac{3^x\left(x-3\right)}{a-b}\times \frac{\left(a^2+ab+b^2\right)}{\left(a^2+ab+b^2\right)} \\ = \underset{x\to 3}{\lim }\frac{3^x\left(x-3\right)\left(a^2+ab+b^2\right)}{a^3-b^3} \\ = \underset{x\to 3}{\lim }\frac{3^x\left(x-3\right)\left[{\left(\sqrt[3]{x-2}\right)}^2+\sqrt[3]{x-2}\left(1\right)+1^2\right]}{\left(x-2\right)-1} \\ = \underset{x\to 3}{\lim }\frac{3^x\cancel{\left(x-3\right)}\left[{\left(x-2\right)}^{{\displaystyle \frac{2}{3}}}+{\left(x-2\right)}^{{\displaystyle \frac{1}{3}}}+1\right]}{\cancel{x-3}} \\ \mathrm{ตัด} (x-3) \mathrm{ทั้งเศษและส่วน} \end{gathered}$$
$$\begin{gathered} = \underset{x\to 3}{\lim }{3}^x\left[{\left(x-2\right)}^{\frac{2}{3}}+{\left(x-2\right)}^{\frac{1}{3}}+1\right] \\ = 3^3\left[{\left(3-2\right)}^{\frac{2}{3}}+{\left(3-2\right)}^{\frac{1}{3}}+1\right] \\ = 27\left(1+1+1\right) \\ = 81 \\ \mathrm{ดังนั้น} \underset{x\to 3}{\lim }\frac{3^x·x-3^{\left(x+1\right)}}{\sqrt[3]{x-2}-1} = 81 \end{gathered}$$