If a function is considered rational and the denominator is not zero, the limit can be found by substitution. This can be seen in the example below (which is similar to the example #3 above, but now done in …

Limits are a very powerful tool in mathematics and are used throughout calculus and beyond. The key idea is that a limit is what I like to call a \behavior operator". A limit will tell you the behavior of a …

Limit Theorems Basic Properties of Limits - Let f : A n m and g : A n m with x0 A or a −→ R ⊂ R −→ R ∈ boundary point of A. If lim f(x) = b1 and lim g(x) = b2, then

lim f(x) = L means that f(x) is close to the number L. This is the most common type of limit. lim f(x) = ∞ means that f(x) grows without bound, eventually be-coming bigger than any number you can name. …

Bottom lines: The limit of a sum/difference/product is the sum/difference/product of the limits. For the most part, the limit of a quotient is the quotient of the limits, except when the limit of the denominator …

The good thing about this de nition is that it de nes the limit in terms of the ordinary ideas of subtracting numbers and comparing them with <. It gets rid of the vague and imprecise idea of \approaching" or …

You might expect that there would be a rule that says “the limit of a quotient is the quotient of the limits”. There is — though we have to be careful that the component limits exist, and also that we avoid …