This page helps you learn analytic methods of finding limit of a function and become familiar with elementary theorems.
G. M. Fichtenholz – Integral and differential calculus, vol. 1, PWN Warsaw 1999 (originally in Russian)
Let us define two variables x and y that are defined in sets: X and Y, respectively. We assume that variable x can take any value from X. Then variable y is called function of x in X if there is a rule prescribing each value x from X only one value y (in Y).
where f denotes that rule (which allows make projections of X to Y).
Limit of f(x) function where x = 1, 2, …, n is defined for x → ∞ and it is the limit of the sequence f(1), f(2), … f(n) for n → ∞
Function f(x) has at the point x = x_{0} limit a
if for x tending to x_{0} values of function f(x) are tending to a. Function f(x) taken at the point x_{0} can differ from a and even can have no assigned value at x_{0}.
Function f(x) has the limit A when x tends to x_{0} (or at the point x_{0}) if for any ε > 0 there is such δ > 0 that
where x ≠ x_{0}.
Let us concern another important function property related to above - presented definition, namely, function continuity. Consider a function f(x) that is defined in the space Ω = {x} where x_{0} is the limit point. This point x_{0} belongs to Ω and the function has assigned value at this point f(x_{0}). If
then we say that the function f(x) is continuous at the point x_{0}. Otherwise we say that the function f(x) is discontinuous at the point x_{0}.
If for x tending to x_{0} function f(x) has the finite limit A then for A > p (A < q) and x close enough x_{0} (but different from x_{0}) the function f(x) fulfills
Let us take ε > 0 and such that
we have
From the limit definition we can find such δ that for
we have
It means that
what should be shown.
If you want to exercise some analytical examples of finding function limit go to the page function limit - analytical examples.