Limits

A limit describes the behaviour of a function as its argument approaches either a fixed constant or infinity.
You can use limits to understand functions which can cause division by zero.
You can also use limits to understand series which have infinite values.

Limits describe the value of a function at a certain input in terms of its value at nearby input.
Limits are the easiest way to provide rigorous foundations for calculus.

If the function is continuous (without gaps) then the function value equals the limit.
The value at the limit (or even whether it exists at all) is irrelevant when you calculating a limit.

One Sided

These work the same as two-sided limits except that x approaches from just left or the right.
These types of limits are used in the formal definitions of limits.
To indicate one sided you just put a little superscript sign on the x number.
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Let f be a function and let a be a real number, then
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exists if and only if
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In real terms conditions (3) is all you need to check.

Two Sided

Limit Definitions

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Approaching Infinity

Limits can show us what happens to an expression when x increases towards infinity