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Did You Know?

As she does so, she notices that her altitude approaches L.

If both of these limits are equal to L then this can be referred to as the limit of f(x) at p.

and the relative limit for each of those sets exist and is the equal to, say, L, then the limit exists for the point x and is equal to L.In particular, if f is real-valued, then the limit of f at p is L if and only if both the right-handed limit and left-handed limit of f at p exist and are equal to L.The function f is continuous at p if and only if the limit of f(x) as x approaches p exists and is equal to f(p).

Although the function (sin x)/x is not defined at zero, as x becomes closer and closer to zero, (sin x)/x becomes arbitrarily close to 1.

Such a view is fundamental in the field of general topology, where limits and continuity at a point are defined in terms of special families of subsets, called filters, or generalized sequences known as nets.Alternatively, the requirement that Y be a Hausdorff space can be relaxed to the assumption that Y be a general topological space, but then the limit of a function will not be unique.