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We can give a more intuitive definition than the formal definition, which is called the \delta and \epsilon definition. Given an \epsilon, no matter how small it is, we can always find a \delta that |f(x_1+\delta)-f(x_1-\delta)| < 2\epsilon. When this is true, we say the limits when x approach x_1 exist. (x_1 is some constant). You can write it as follows. $$ \lim_{x\to x_1}f(x)=L $$

Note that this limit exists on both sides of the point. There are other limits, which may only exist on the left or on the right. They can be noted as follows:

\lim_{x\to x_1^{-}}f(x) \\ \lim_{x\to x_1^{+}}f(x)