# Understanding "x Approaches c" Notation

- Author:
- Ken Schwartz

Choose any function , and choose any value . Drag the green point " -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.

**x**" anywhere along theWhen 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 " -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 . Ditto for the "right" side; in this case, is always , and we decrease to make it . In either case, always stays on the same side of as it changes.
The reason we do this from 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 , and move to the , 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 approaches , and as approaches from the approaches ". In math notation this is written "As , , and as , ".

**x**" on the*closer*to the value of*greater*than*closer*to__both__directions is that we might see the value of*right*of*left*toward__,__*left*__,__*right*## New Resources

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