Limit of a function

In calculus, a branch of mathematics, the limit of a function is the behavior of a certain function near a selected input value for that function. Limits are one of the main calculus topics, along with derivatives, integration, and differential equations.
Definition of the limit.
The definition of the limit is as follows:
The notation for the limit above is read as "The limit of formula_1 as formula_3 approaches formula_4 is formula_2", or alternatively, formula_10 as formula_11 (reads "formula_1 tends to formula_2 as formula_3 tends to formula_4"). Informally, this means that we can make formula_1 as close to formula_2 as possible—by making formula_3 sufficiently close to formula_4 from both sides (without making formula_3 equal to formula_4).
Imagine we have a function such as formula_22. When formula_23, formula_1 is undefined, because formula_25. Therefore, on the Cartesian coordinate system, the function formula_22 would have a vertical asymptote at formula_23. In limit notation, this would be written as:
Right and left limits.
For the function formula_22, we can get as close to formula_30 in the formula_3-values as we want, so long as we do "not" make formula_3 equal to formula_30. For instance, we could make x=.00000001 or -.00000001, but never 0. Therefore, we can get formula_1 as close as we want to formula_31, but without reaching it. The left limit is any value that approaches the limit from numbers less than the number, and the right limit is any value that approaches the limit from number greater than the limit number. For instance, in the function formula_22, since the limit for formula_3 is 0, if formula_42, it approaches the limit from the right. If we instead choose -1, we say it approaches the limit from the left.