Question

Given epsilon = 1, find the largest value of delta graphically and algebraically

Answer 1

\[\lim_{x \rightarrow 10} = \sqrt{19-x}\]

Answer 2

I assume you meant to write \(\lim\limits_{x\to10}\sqrt{19-x}=3\) ?

You're looking for \(\delta\) such that \(|x-10|<\delta\) returns \(|\sqrt{19-x}-3|<1\). Solve this inequality for \(x\):
\[\begin{align*}
-1&<\sqrt{19-x}-3<1\\[1ex]
2&<\sqrt{19-x}<4\\[1ex]
4&<19-x<16\\[1ex]
-15&<-x<-3\\[1ex]
3&<x<15
\end{align*}\]From here, you want to extract information about \(|x-10|\), which can be done by further solving explicitly for \(x-10\):
\[3<x<15\implies -7<x-10<5\]which means the maximum distance in absolute value that \(x\) can be from \(10\) that guarantees \(|\sqrt{19-x}-3|<1\) must be \(\delta=5\).