Question

Find the set of δ values that satisfy the formal definition of lim(11/g(x))=2.2 as x approaches 4 when given the value ε = 0.5 and g(x) = x2 – x – 7, showing all work.

Answer 1

The epsilon-delta definition of the limit says \[\lim_{x \rightarrow a} f(x) = L \]means \[\forall \epsilon > 0 : \exists \delta > 0 : \forall x \in U : |x-a|<\delta \Rightarrow |f(x)-L|<\epsilon,\]where \(U\) is the domain of \(f\), so,\[\left| 2.2 - \frac{11}{x^2-x-7} \right| < \epsilon. \]For sufficiently small \(\epsilon\), this implies\[\color{red}{\frac{1}{2}-\frac{1}{2}\sqrt{\frac{145 \epsilon - 539}{5 \epsilon-11}} < x < \frac{1}{2}-\frac{1}{2}\sqrt{\frac{145 \epsilon + 539}{5 \epsilon +11}}}.\]However, we know \(a=4\) and assumed \[|x-a|=|x-4|<\delta \Rightarrow \color{red}{4 - \delta< x < \delta + 4 } \]Both of the expressions I indicated in red must simultaneously hold. Does that give you enough of a hint?

Answer 2

It does. Thank you very much for your help. I believe I understand it much better now :)

Answer 3

Great :-)