Choose any function [math]f[/math], and choose any value [math]c[/math]. Drag the green point "[b][color=#38761d]x[/color][/b]" anywhere along the [math]x[/math]-axis. When you click the "Let x Approach c" button, [math]x[/math] moves halfway to [math]c[/math], from whichever side it happens to be on. The text in the right-hand pane tells you what's going on.

When we write [math]x\longrightarrow c[/math], we say "[math]x[/math] approaches [math]c[/math]". This just means that we are imagining the value of [math]x[/math] changing and becoming closer to the value of some number [math]c[/math]. The value of "[math]c[/math]" is some [math]x[/math]-value, near which we want to examine the function's behavior. So we picture a moving point "[b][color=#6aa84f]x[/color][/b]" on the [math]x[/math]-axis "sneaking up" on [math]c[/math], and we observe what the function's value at [math]x[/math], [math]y=f\left(x\right)[/math], is doing as this happens.[br][br]Notice that [math]x[/math] could be located on either side of [math]c[/math], left or right. When we write [math]x\longrightarrow c-[/math], we say "[math]x[/math] approaches [math]c[/math] from the left". This means that we keep the value of [math]x[/math] less than the value of [math]c[/math], but we are increasing the value of [math]x[/math] to make it [i]closer[/i] to the value of [math]c[/math]. Ditto for the "right" side; in this case, [math]x[/math] is always [i]greater[/i] than [math]c[/math], and we decrease [math]x[/math] to make it [i]closer[/i] to [math]c[/math]. In either case, [math]x[/math] always stays on the same side of [math]c[/math] as it changes.[br][br]The reason we do this from [u]both[/u] directions is that we might see the value of [math]y[/math] doing different things on the two different sides of [math]c[/math]. In the piecewise function that the app defaults to, [math]y[/math] increases in value towards [math]y=3[/math], as [math]x[/math] moves to the right toward [math]c[/math] ([math]c[/math] has a value of [math]2[/math]). But if we instead start [math]x[/math] somewhere to the [i]right[/i] of [math]c[/math], and move [math]x[/math] to the [i]left[/i] toward [math]c[/math], we see [math]y[/math] changing value toward [math]y=-1[/math] instead of [math]y=3[/math]. [color=#cc0000]The one-sided behavior of [/color][math]f[/math][color=#cc0000] is different to the left of [/color][math]c[/math][color=#cc0000] than it is to the right of [/color][math]c[/math]. So, we would say "as [math]x[/math] approaches [math]2[/math] from the [u][i]left[/i][/u], [math]y[/math] approaches [math]3[/math], and as [math]x[/math] approaches [math]2[/math] from the [u][i]right[/i][/u], [math]y[/math] approaches [math]-1[/math]". In math notation this is written "As [math]x\longrightarrow c-[/math], [math]y\longrightarrow3[/math], and as [math]x\longrightarrow c+[/math], [math]y\longrightarrow-1[/math]".