So, essentially, I am stuck on this problem. Note: I must solve problem using the definition I listed in question. I will provide Picture of my work down below and the question.

You are correct sofar. I will not do this for you but will gladly check your work.

You have absolute control over \(\displaystyle \delta\) but \(\displaystyle \varepsilon \) is a fixed positive number.

Look at what you have done:

\(\displaystyle \begin{align*}|f(x)-f(1)|&=\left|\frac{x-1}{x(\sqrt{x}+1)}\right| \\&=\frac{|x-1|}{|x(\sqrt{x}+1)|}\\&\le\delta\frac{1}{|x(\sqrt{x}+1)|} \end{align*}\)

Can you adjust \(\displaystyle \delta\) so as to make \(\displaystyle \frac{\delta}{|x(\sqrt{x}+1)|}<\varepsilon~~?\)