Question

In Problems 45-60, determine whether $f$is continuous at $c$.

$$\begin{gathered} f\left(x\right)=x^3-3x^2+2x-6, \\ c=2 \end{gathered}$$

Short answer

The function is continuous at$c$

Step-by-step solution

Given information

Given function

$$\begin{gathered} f\left(x\right)=x^3-3x^2+2x-6 \\ c=2 \end{gathered}$$

Check whether the function is defined at c

Checking, we get

$$\begin{gathered} f\left(2\right)=2^3-3(2{)}^2+2(2)-6 \\ f(2)=8-3(4)+2(2)-6 \\ f(2)=8-12+4-6 \\ f(2)=-6 \end{gathered}$$

So it is defined at$c$

Check the left hand limits and right hand limits

We have,

$$\begin{gathered} \underset{x\to 2^{-}}{\lim }f\left(x\right)=2^3-3(2{)}^2+2(2)-6 \\ \underset{x\to 2^{-}}{\lim }f(x)=8-3(4)+2(2)-6 \\ =-6 \end{gathered}$$

Again,

$$\begin{gathered} \underset{x\to 2^{+}}{\lim }f\left(x\right)=\underset{x\to 2^{+}}{\lim }{2}^3-3(2{)}^2+2(2)-6 \\ \underset{x\to 2^{+}}{\lim }f(x)=\underset{x\to 2^{+}}{\lim }8-3(4)+2(2)-6 \\ \underset{x\to 2^{+}}{\lim }f(x)=-6 \end{gathered}$$

Since $\underset{x\to c}{\lim }f\left(x\right)=f\left(c\right)$so it is continuous at$c$