proof of limit rule of product

Let f and g be real ( or complex functions having the limits


Then also the limit limxx0f(x)g(x) exists and equals FG.

Proof.  Let ε be any positive number.  The assumptionsPlanetmathPlanetmath imply the existence of the positive numbers δ1,δ2,δ3 such that

|f(x)-F|<ε2(1+|G|)when  0<|x-x0|<δ1 (1)
|g(x)-G|<ε2(1+|F|)when  0<|x-x0|<δ2, (2)
|g(x)-G|<1when  0<|x-x0|<δ3. (3)

According to the condition (3) we see that

|g(x)|=|g(x)-G+G||g(x)-G|+|G|<1+|G|when  0<|x-x0|<δ3.

Supposing then that  0<|x-x0|<min{δ1,δ2,δ3}  and using (1) and (2) we obtain

|f(x)g(x)-FG| =|f(x)g(x)-Fg(x)+Fg(x)-FG|

This settles the proof.

Title proof of limit rule of product
Canonical name ProofOfLimitRuleOfProduct
Date of creation 2013-03-22 17:52:22
Last modified on 2013-03-22 17:52:22
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Proof
Classification msc 30A99
Classification msc 26A06
Related topic ProductOfFunctions
Related topic TriangleInequality
Related topic ProductAndQuotientOfFunctionsSum