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## Homework Statement

Suppose the functions f and g have the following property: for all ε > 0 and all x,

If [itex]0 < \left| x-2 \right| < \sin^{2} \left( \frac{\varepsilon^{2}}{9} \right) + \varepsilon[/itex], then [itex]\left| f(x) - 2 \right| < \varepsilon[/itex].

If [itex]0 < \left| x - 2 \right| < \varepsilon^{2}[/itex] then [itex]\left| g(x) - 4 \right| < \varepsilon[/itex].

For each [itex]\varepsilon > 0[/itex] find a [itex]\delta > 0[/itex] such that for all x:

ii) if [itex]0 < \left| x-2 \right| < \delta[/itex], then [itex]\left| f(x)g(x) - 8 \right| < \varepsilon[/itex].

## Homework Equations

## The Attempt at a Solution

The solution book says:

We need: [itex]\left| f(x) - 2 \right| < min \left( 1, \frac{\varepsilon}{2 (\left| 4 \right| + 1)} \right) [/itex] and [itex]\left| g(x) - 4 \right| < \frac{\varepsilon} {2 (\left| 2 \right|) + 1}[/itex].

My question is: How did they get these fractions in the solution? I've multiplied [itex]\left| f(x) - 2 \right|[/itex] and [itex]\left| g(x) - 4 \right| [/itex] to get [itex]\left| f(x)g(x) - 4f(x) - 2g(x) + 8 \right|[/itex]. Am I going in the right direction?