I am stuck at the denominator of the limit proving lim x→∞ 1/(x3-2x-1) = 0

[harieche]

I am stuck at the denominator of the limit proving lim x→∞ 1/(x3-2x-1) = 0

The question is prove by the definition that lim x→∞ 1/(x³-2x-1) = 0
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For now, I have done this:

If for every epsilon > 0 , there exists x₁ such that

| 1/(x³-2x-1) - 0 | < epsilon , wherever x₁< x

| 1/(x³-2x-1) | < epsilon

| x³-2x-1 | > 1/epsilon


Then I don't know what to do next, can anyone help me please? Thank you

 

[Ishuda]

The question is prove by the definition that lim x→∞ 1/(x³-2x-1) = 0
_______________________________________________________

For now, I have done this:

If for every epsilon > 0 , there exists x₁ such that

| 1/(x³-2x-1) - 0 | < epsilon , wherever x₁< x

| 1/(x³-2x-1) | < epsilon

| x³-2x-1 | > 1/epsilon


Then I don't know what to do next, can anyone help me please? Thank you

This looks like one of those questions you may want to have a segmented answer for, i.e. maybe something like
\(\displaystyle x_1\, =\, max\left(100,\, \dfrac{2}{\epsilon^\dfrac{1}{3}}\right)\)

To obtain the actual numbers, you need to find (or at least limit the size of x for) the zero's for x³-2x-1 and say the relative maximas of 1-2/x²-1/x³