# 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) 