The definition of a limit: The expression is an abbreviation for: the value of the single-variable function approaches as approaches the value . More formally, this means that can be made arbitrarily close to by making sufficiently close to , or in precise mathematical terms, for each real , there exists a such that . In other words, the inequalities state that for all except within of , is within of .
This definition extends to multivariable functions as distances are measured with the Euclidean metric.
In the figure, the horizontal planes represent the bounds on and the cylinder is . No matter what is given, a is found (represented by the changing radius of the cylinder) so that all points on the surface inside the cylinder are between the two planes.
For the limit of a multivariable function, consider the two-variable function . (Note that the following extends to functions of more than just two variables, but for the sake of simplicity, two-variable functions are discussed.) The same limit definition applies here as in the one-variable case, but because the domain of the function is now defined by two variables, distance is measured as , all pairs within of are considered, and should be within of for all such pairs . As an example, here is a proof that the limit of is 10 as . Claim: for a given , choosing satisfies the appropriate conditions for the definition of a limit: (the given condition) reduces to , which implies that and .
Now, by the triangle inequality, and . If , , and if , . Thus by the choice of , , and because is arbitrary, an appropriate can be found for any value of ; hence the limit is 10.