Question

Does $\underset{x\to 0}{\lim }f\left(x\right)$exist? If it does, what is it?

Short answer

The limit$\underset{x\to 0}{\lim }f\left(x\right)$does not exist.

Step-by-step solution

Step 1. Graph the function f(x)

We have been given$\underset{x\to 0}{\lim }f\left(x\right)$.

Now let us graph the function with the help of a graphing utility.

Step 2. Determine if the limit exists. 

For the limit at $0$to exist, then$\underset{x\to 0^{-}}{\lim }f\left(x\right)=\underset{x\to 0^{+}}{\lim }f\left(x\right)$.

Let us consider the behavior of $f\left(x\right)$near$x=0$.

When we consider the limit coming from the left side, or$\underset{x\to 0^{-}}{\lim }f\left(x\right)$, the function approaches a value of $4$.

That is$\underset{x\to 0^{-}}{\lim }f\left(x\right)=4$

When we consider the limit coming from the right side, or$\underset{x\to 0^{+}}{\lim }f\left(x\right)$, the function approaches a value of $1$.

That islocalid="1647086140473" $\underset{x\to 0^{+}}{\lim }f\left(x\right)=1$

Since$\underset{x\to 0^{-}}{\lim }f\left(x\right)\ne \underset{x\to 0^{+}}{\lim }f\left(x\right)$, the limit$\underset{x\to 0}{\lim }f\left(x\right)$does not exist.