Show that the statement limx→0x3=0 is correct using the ε, δ definition of limit.
Based from the defintion,
xif 0<|x−a|<δ then |f(x)−L|<εxif 0<|x−0|<δ then |x3−0|<ε
That is,x if 0<|x|<δ then |x3|<ε
Or, taking the cube root of both sides of the inequality |x3|<ε , we get...
if 0<|x|<δ then |x|<3√ε
The statement suggests that we should choose δ=3√ε
By proving that the assumed value of δ=3√ε will fit the definition...
if 0<|x|<δ then, x|x3|<δ3=(3√ε)3=ε
Thus, xif 0<|x|<δ then |x3|<εTherefore, by the definition of a limitxlimx→0x3=0
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