Suppose the function $f$ and $g$ have the following property: $\forall$$\epsilon > 0$ and $\forall x$
if $0 < |x-a|<\sin^{2}(\frac{\epsilon^{2}}{9})+\epsilon$, then $|f(x)-2|<\epsilon$
if $0 < |x-a|< \epsilon^{2}$, then $|g(x) – 4|<\epsilon$
For each $\epsilon > 0$ find a $\delta > 0$ such that, $\forall x$
(i) if $0<|x-a|<\delta$, then $|f(x) + g(x) -6|<\epsilon$
Best Answer
You can take $\delta = \min\{\sin^2(\epsilon^2/36) + \epsilon/2,\epsilon^2/4 \}$. Because then $|f(x)+g(x)-6| \leq |f(x)-2|+|g(x)-4| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$.