The equation for this problem is would be 3 + (1/(x-4)^2)

This is the problem:

The limit of r(x) as x approaches infinity is 3. How large would you have to keep x in order for r(x) to be within 0.01 unit of 3? How large would you have to keep x in order for r(x) to be within epsilon units of 3 where epsilon is a small positive number.

Please can someone help me solve it step by step.

2. Originally Posted by Plasma540
The equation for this problem is would be 3 + (1/(x-4)^2)

This is the problem:

The limit of r(x) as x approaches infinity is 3. How large would you have to keep x in order for r(x) to be within 0.01 unit of 3? How large would you have to keep x in order for r(x) to be within epsilon units of 3 where epsilon is a small positive number.

Please can someone help me solve it step by step.
You want $\displaystyle \left(3+\frac{1}{(x-4)^2}\right)-3<\epsilon$. So,

$\displaystyle \left(3+\frac{1}{(x-4)^2}\right)-3=\frac{1}{(x-4)^2}<\epsilon \implies (x-4)^2>\frac{1}{\epsilon}$ $\displaystyle \implies x-4>\sqrt{\frac{1}{\epsilon}}\implies x>4+\sqrt{\frac{1}{\epsilon}}$.

For the first part, let $\displaystyle \epsilon=0.01$ and solve for $\displaystyle x$. You should get $\displaystyle x>14$.