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Understanding "x Approaches c" Notation

Choose any function , and choose any value . Drag the green point "x" anywhere along the -axis. When you click the "Let x Approach c" button, moves halfway to , from whichever side it happens to be on. The text in the right-hand pane tells you what's going on.
When we write , we say " approaches ". This just means that we are imagining the value of changing and becoming closer to the value of some number . The value of "" is some -value, near which we want to examine the function's behavior. So we picture a moving point "x" on the -axis "sneaking up" on , and we observe what the function's value at , , is doing as this happens. Notice that could be located on either side of , left or right. When we write , we say " approaches from the left". This means that we keep the value of less than the value of , but we are increasing the value of to make it closer to the value of . Ditto for the "right" side; in this case, is always greater than , and we decrease to make it closer to . In either case, always stays on the same side of as it changes. The reason we do this from both directions is that we might see the value of doing different things on the two different sides of . In the piecewise function that the app defaults to, increases in value towards , as moves to the right toward ( has a value of ). But if we instead start somewhere to the right of , and move to the left toward , we see changing value toward instead of . The one-sided behavior of is different to the left of than it is to the right of . So, we would say "as approaches from the left, approaches , and as approaches from the right, approaches ". In math notation this is written "As , , and as , ".