Understanding Left-Hand Limits and Right-Hand Limits
We can approach the input of a function from either side of a value—from the left or the right. The table below shows the values of
[latex]f\left(x\right)=x+1,x\ne 7[/latex]
as described earlier.
Values described as “from the left” are less than the input value 7 and would therefore appear to the left of the value on a number line. The input values that approach 7 from the left are [latex]6.9[/latex], [latex]6.99[/latex], and [latex]6.999[/latex]. The corresponding outputs are [latex]7.9,7.99[/latex], and [latex]7.999[/latex]. These values are getting closer to 8. The limit of values of [latex]f\left(x\right)[/latex] as [latex]x[/latex] approaches from the left is known as the left-hand limit. For this function, 8 is the left-hand limit of the function [latex]f\left(x\right)=x+1,x\ne 7[/latex] as [latex]x[/latex] approaches 7.
Values described as “from the right” are greater than the input value 7 and would therefore appear to the right of the value on a number line. The input values that approach 7 from the right are [latex]7.1[/latex], [latex]7.01[/latex], and [latex]7.001[/latex]. The corresponding outputs are [latex]8.1[/latex], [latex]8.01[/latex], and [latex]8.001[/latex]. These values are getting closer to 8. The limit of values of [latex]f\left(x\right)[/latex] as [latex]x[/latex] approaches from the right is known as the right-hand limit. For this function, 8 is also the right-hand limit of the function [latex]f\left(x\right)=x+1,x\ne 7[/latex] as [latex]x[/latex] approaches 7.
We can get the output of the function within a distance of 0.1 from 8 by using an input within a distance of 0.1 from 7. In other words, we need an input [latex]x[/latex] within the interval [latex]6.9[latex]\underset{x\to {7}^{-}}{\mathrm{lim}}f\left(x\right)=8[/latex].
The left- and right-hand limits are the same for this function.
one-sided limits
The left-hand limit of a function [latex]f\left(x\right)[/latex] as [latex]x[/latex] approaches [latex]a[/latex] from the left is equal to [latex]L[/latex], denoted by
The values of [latex]f\left(x\right)[/latex] can get as close to the limit [latex]L[/latex] as we like by taking values of [latex]x[/latex] sufficiently close to [latex]a[/latex] such that [latex]xright-hand limit of a function [latex]f\left(x\right)[/latex], as [latex]x[/latex] approaches [latex]a[/latex] from the right, is equal to [latex]L[/latex], denoted by
The values of [latex]f\left(x\right)[/latex] can get as close to the limit [latex]L[/latex] as we like by taking values of [latex]x[/latex] sufficiently close to [latex]a[/latex] but greater than [latex]a[/latex]. Both [latex]a[/latex] and [latex]L[/latex] are real numbers.
Understanding Two-Sided Limits
In the previous example, the left-hand limit and right-hand limit as [latex]x[/latex] approaches [latex]a[/latex] are equal. If the left- and right-hand limits are equal, we say that the function [latex]f\left(x\right)[/latex] has a two-sided limit as [latex]x[/latex] approaches [latex]a[/latex]. More commonly, we simply refer to a two-sided limit as a limit. If the left-hand limit does not equal the right-hand limit, or if one of them does not exist, we say the limit does not exist.
two-sided limits
The limit of a function [latex]f\left(x\right)[/latex], as [latex]x[/latex] approaches [latex]a[/latex], is equal to [latex]L[/latex], that is,
In other words, the left-hand limit of a function [latex]f\left(x\right)[/latex] as [latex]x[/latex] approaches [latex]a[/latex] is equal to the right-hand limit of the same function as [latex]x[/latex] approaches [latex]a[/latex]. If such a limit exists, we refer to the limit as a two-sided limit. Otherwise we say the limit does not exist.
