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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)
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