6 August 30, 2022

Limit from the left: Let [latex]𝑓(𝑥)[/latex] be a function defined at all values in an open interval of the form [latex](c, a)[/latex], and let [latex]L[/latex] be a real number. If the values of the function [latex]𝑓(𝑥)[/latex] approach the real number [latex]L[/latex] as the values of [latex]x[/latex] (where [latex]𝑥<𝑎[/latex]) approach the number [latex]a[/latex], then we say that L is the limit of [latex]𝑓(𝑥)[/latex] as [latex]x[/latex] approaches a from the left. Symbolically, we express this idea as When using set-builder notation to describe a subset of all real numbers, denoted [latex]ℝ,[/latex] we write

[latex]\{x\vert x\;has\;some\;property\}.[/latex]

We read this as the set of real numbers [latex]x[/latex] such that [latex]x[/latex] has some property. For example, if we were interested in the set of real numbers that are greater than one but less than five, we could denote this set using set-builder notation by writing

[latex]\{x\vert 1 \lt x \lt 5\}[/latex]

A set such as this, which contains all numbers greater than [latex]a[/latex] and less than [latex]b,[/latex] can also be denoted using the interval notation [latex](a,b).[/latex] Therefore,

[latex](1,5)=\{x\vert 1 \lt x \lt 5\}[/latex]

The numbers [latex]1[/latex] and [latex]5[/latex] are called the endpoints of this set. If we want to consider the set that includes the endpoints, we would denote this set by writing

[latex][1,5]=\{x\vert 1 \leq x \leq 5\}[/latex]

We can use similar notation if we want to include one of the endpoints, but not the other. To denote the set of nonnegative real numbers, we would use the set-builder notation

[latex]\{x\vert 0 \leq x\}[/latex]