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.

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: 2x2 + 4x − 1=0x→∞3x3 + 1lim So we have give &gt; 0 and we have to find N ( ) such that| 2x2 + 4x − 12x2 + 4x − 1− 0| = ||&lt;3+13x3x3 + 1 for x &gt; N ( )....