Question

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

role="math" localid="1647000158813" $$\begin{gathered} f\left(x\right)=3x^2-6x+5 \\ c=-3 \end{gathered}$$

Short answer

The function is continuous at$c$

Step-by-step solution

Given information

Given the function$f\left(x\right)=3x^2-6x+5$

Check whether the function is defined at c

Checking, we get

$$\begin{gathered} f(-3)=3(-3{)}^2-6(-3)+5 \\ f(-3)=3(9)-6(-3)+5 \\ f(-3)=27+18+5 \\ f(-3)=50 \end{gathered}$$

So the function is defined

Check the left hand limits and right hand limits

Checking, we get

$$\begin{gathered} \underset{x\to 3^{-}}{\lim }f\left(x\right)\underset{x\to 3^{-}}{\lim }3x^2+6x+5 \\ \underset{x\to 3^{-}}{\lim }f\left(x\right)\underset{x\to 3^{-}}{\lim }3{\left(3\right)}^2+6\left(3\right)+5 \\ \underset{x\to 3^{-}}{\lim }f\left(x\right)=50 \end{gathered}$$

$$\begin{gathered} \underset{x\to -3^{+}}{\lim }f\left(x\right)=\underset{x\to -3^{+}}{\lim }3x^2-6x+5 \\ \underset{x\to -3^{+}}{\lim }f\left(x\right)=\underset{x\to -3^{+}}{\lim }3{(-3)}^2-6(-3)+5 \\ \underset{x\to -3^{+}}{\lim }f\left(x\right)=50 \end{gathered}$$

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