Question

In Problems 1– 6, find each limit

$\underset{x\to 5}{\lim }\left[\left(3x\right)(x-2{)}^2\right]$

Short answer

The value is$135$

Step-by-step solution

Step 1. Rewriting the limit

The function can be rewritten as,

$$\begin{gathered} \left(3x\right)(x-2{)}^2=3x\left(x^2-4x+4\right) \\ =3x^3-12x^2+12x \end{gathered}$$

Therefore,

$\underset{x\to 5}{\lim }\left[\left(3x\right)(x-2{)}^2\right]=\underset{x\to 5}{\lim }\left(3x^3-12x^2+12x\right)$

Step 2. Simplifying the limit

$$\begin{gathered} \underset{x\to 5}{\lim }\left[\left(3x\right)(x-2{)}^2\right]=\underset{x\to 5}{\lim }\left(3x^3-12x^2+12x\right) \\ \underset{x\to 5}{\lim }\left(3x^3-12x^2+12x\right)=\underset{x\to 5}{\lim }3x^3-\underset{x\to 5}{\lim }12x^2+\underset{x\to 5}{\lim }12x \\ =3(5{)}^3-12(5{)}^2+12\left(5\right) \\ =375-300+60 \\ =135 \end{gathered}$$