Calculus Limit Question

Question

If lim x approachess 0 f(x)/Ax^4+Bx^3 =C.where A,B,C are constans.find lim x approachess 0 f(x)/x^2

Answer 1

Alright

I suppose You have f(x)/(Ax^4+Bx^3). In this case

 

If Lim_x→0f(x)/(Ax^4+Bx^3) =C mean that

∃ε>0 such that 

|f(x)/(Ax^4+Bx^3)-C|<ε,

 

for all δ>0 such that |x-x₀|<δ

From 

|f(x)/(Ax^4+Bx^3)-C|<ε  follows that

 

C-ε<f(x)/(Ax^4+Bx^3)< C+ε

 

(C-ε)(Ax^2+Bx)<f(x)/x^2< (C+ε)(Ax^2+Bx)

 

- ε(Ax^2+Bx)+C(Ax^2+Bx) <f(x)/x^2<ε(Ax^2+Bx)+C(Ax^2+Bx)

 

- ε(Ax^2+Bx) <f(x)/x^2-C(Ax^2+Bx)<ε(Ax^2+Bx) (*)

 

For x→0, 

 

- ε₁ <Lim _x→0 f(x)/x^2<ε₁

This shows  that  Lim _x→0 f(x)/x^2=0.

 

I hope I answered you question.