Courses in Math. Function Limit - Definition & Theorems.

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)

Function. Definition.

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).


y = f(x)

where f denotes that rule (which allows make projections of X to Y).

Functions of integer argument. Limit.

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 → ∞

lim f(x) for x → ∞ = lim f(n) for n → ∞ = a

Functions in connected space. Limit.

Function f(x) has at the point x = x0 limit a

lim f(x) for x → x0 = a

if for x tending to x0 values of function f(x) are tending to a. Function f(x) taken at the point x0 can differ from a and even can have no assigned value at x0.

Limit - strict formulation.

Function f(x) has the limit A when x tends to x0 (or at the point x0) if for any ε > 0 there is such δ > 0 that

|f(x) - A| < ε only if |x - x0| < δ

where x ≠ x0.

Function continuity at point.

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 x0 is the limit point. This point x0 belongs to Ω and the function has assigned value at this point f(x0). If

lim f(x) when x → x0 = f(x0)

then we say that the function f(x) is continuous at the point x0. Otherwise we say that the function f(x) is discontinuous at the point x0.

Function limit - theorem.

If for x tending to x0 function f(x) has the finite limit A then for A > p (A < q) and x close enough x0 (but different from x0) the function f(x) fulfills

f(x) > p (f(x) < q).


Let us take ε > 0 and such that

ε < A - p (ε < q - A)

we have

A - ε > p (A + ε < q).

From the limit definition we can find such δ that for

|x - a| < δ

we have

A - ε < f(x) < A + ε.

It means that

f(x) > A - ε > p
f(x) < A + ε < q.

what should be shown.

de l'Hospital's rule.

  1. Symbols of ∞/∞

Function Limit - Analytical Examples.

If you want to exercise some analytical examples of finding function limit go to the page function limit - analytical examples.