Prove that $\lim\limits_{x\rightarrow 4} \sqrt{2x+7} = \sqrt{15}$. Prove that $\displaystyle \lim_{x\rightarrow 4} \sqrt{2x+7} = \sqrt{15}$ using the epsilon-delta definition.
This is what I have, but I know my delta value is incorrect. My professor said that it was the right path but my delta is incorrect.
Proof: Let $\varepsilon>0$. Choose $\delta$ such that $0<\delta<\min(\varepsilon,1)$. This means that both $\delta<1$ and $\delta<\varepsilon$. Let $x\in\mathbb{R}$ such that $0<|2x-8|<\delta$. Since $\delta<1$, we have  
$$\begin{array}{cccccc}
 &-1 &< & 2x-8 & < & 1\\
\Rightarrow & 7 &<& 2x &<& 9 \\
\Rightarrow & 7/2 & < & x & < & 9/2
 \end{array}$$
Since $7/2<x<9/2$, 
$$\begin{array}{cccccc}
 &7/2 & < & x & < & 9/2\\
\Rightarrow & 7 &<& 2x &<& 9 \\
\Rightarrow & 7+7 & < & 2x+7 & < & 9+7 \\
\Rightarrow & \sqrt{14} & < & \sqrt{2x+7} & < & \sqrt{16} \\
\Rightarrow & \sqrt{14} + \sqrt{15} & < & \sqrt{2x+7}+\sqrt{15} & < & \sqrt{16}+\sqrt{15}\\
\Rightarrow & \displaystyle \frac{1}{\sqrt{14} + \sqrt{15}} & > & \displaystyle \frac{1}{\sqrt{2x+7}+\sqrt{15}} & > & \displaystyle \frac{1}{\sqrt{16} + \sqrt{15}}\\
 \end{array}$$
This implies $$\left|\frac{1}{\sqrt{2x+7}+\sqrt{15}}\right|< \frac{1}{\sqrt{14} + \sqrt{15}}<1.$$
Therefore, 
$$\begin{align*}
\left|\sqrt{2x+7}-\sqrt{15}\right|
&= \left|\left(\sqrt{2x+7}-\sqrt{15}\right) \cdot \left(\frac{\sqrt{2x+7}+\sqrt{15}}{\sqrt{2x+7}+\sqrt{15}}\right)\right| \\
&= \left|2x+7-15\right| \cdot \left| \frac{1}{\sqrt{2x+7}+\sqrt{15}}\right|\\
&=\left|2x-8\right|\cdot \left|\frac{1}{\sqrt{2x+7}+\sqrt{15}}\right|\\
&< \delta \cdot 1 \\
&< \varepsilon \cdot 1\\
\end{align*}$$
Thus, $|\sqrt{2x+7}-\sqrt{15}|<\varepsilon$. So, $\displaystyle \lim_{x\rightarrow 4} \sqrt{2x+7} = \sqrt{15}$.

Denote by $D$ the domain of $f$, where $f(x) = \sqrt{2x + 7}$.
Let $\varepsilon > 0$. Choose $\delta = (\sqrt{15}/2) \varepsilon$. Let $x \in D$. If $0 < |x-4| < \delta$, then $$\begin{aligned}[t] |f(x) - \sqrt{15}| = \biggl| \dfrac{(f(x) - \sqrt{15}\,)(f(x) + \sqrt{15}\,)}{f(x) + \sqrt{15}}\biggr| &= \dfrac{1}{f(x) + \sqrt{15}} \cdot |(2x+7)-15| \\ &= \dfrac{2}{f(x) + \sqrt{15}} \cdot |x-4| \\ &< \dfrac{2}{\sqrt{15}} \cdot \delta \\ &= \dfrac{2}{\sqrt{15}} \cdot \dfrac{\varepsilon \sqrt{15}}{2} = \varepsilon.\end{aligned} $$
Therefore, $\lim_{x \to 4} \sqrt{2x+7} = \sqrt{15}$.
Remember that if you are trying to prove $\lim_{x \to c} f(x) = L$ using the $\varepsilon$-$\delta$ definition, the structure of your proof should be like so:

"Let $\varepsilon > 0$."
        [A choice for $\delta$ goes here.]
              "Let $x \in \operatorname{dom}(f)$."
                    "Suppose $0 < |x-c| < \delta$."
                    [Proof that $|f(x) - L| < \varepsilon$ goes here.]
  "Thus, $\lim_{x \to c} f(x) = L$."

