**One-sided limit**

In examining which number the value of a function approaches, when a variable approaches this point, we can approach it from both sides of the number.

The figure shows a graph of a function. The graph of a function consists of two parts: a half-line on the left and a curve on the right.

If we approach the value of *x = 4 *along the half-line, we end up with the value of the function* 5*. As we approach along the curve from the right to* x = 4*, we end up with the value of the function *4.*

The left limit of the function is thus *5* and the right limit is *4.*

There is no limit value for this function in *4*, because the one-sided limit values are different.

This is a piecewise function.

The function *f* has a limit value at *a*, if

In the notation, a minus sign means the limit value on the left, i.e. point* (a)* is approached from the negative side, and a plus sign means the limit value on the right, i.e. point* (a)* is approached from the positive side.

**Example 1**

Define the one-sided limits of the function *f*

Solution

Find the limit values. When approaching from the left, the function is defined by a straight line and from the right by a parabola.

The limit value on the left

the limit value on the right

The one-sided limit values are different, so the function has no limit value when *x* approaches* 4*. This is the same function that has a graph at the beginning of the section.

**Example 2**

Find the limit value of the function *g* when *x* approaches *3*

Solution

A function has a limit value if its one-sided limit values are equal.

The limit value on the left

The limit value on the right

The one-sided limit values are equal, so that the function *g* has a limit value at point *3*

Graph of *g(x)*