### Use The Following Definition To Prove The Indicated Limit Let F Be A Function Wi

Use the following definition to prove the indicated limit.Let f be a function with D(f) such that D(f) ∩ (M, ∞) ≠ the empty set. Then we say the limit of f as x goes to infinity is L provided that, given any ε>0, there is a number N(ε) such that if x> N(ε) and xЄD(f), then │f(x) -L│< ε.Prove limx–>∞ (2x^2+4x-1)/(3x^3+1) =0I know we need to find N(ε). So we look at │f(x) -L│< ε and manipulate for x > N(ε) and then that is our N(ε).But I don’t know how to manipulate the equation: │(2x^2+4x-1)/(3x^3+1) │< ε to x > N(ε)Please help.

We have to prove: 2×2 + 4x − 1=0x→∞3×3 + 1lim So we have give > 0 and we have to ﬁnd N ( ) such that| 2×2 + 4x − 12×2 + 4x − 1− 0| = ||<3+13x3x3 + 1 for x > N ( )….