<s>[INST] 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, $\displaystyle \lim_{x\rightarrow 4} \sqrt{2x+7} = \sqrt{15}$.
 [/INST] Hint:
$$
\sqrt{2x+7} - \sqrt{15} = \frac {(\sqrt{2x+7} - \sqrt{15} )(\sqrt{2x+7} + \sqrt{15} )}{\sqrt{2x+7} + \sqrt{15} } = \frac{2x-8}{\sqrt{2x+7} + \sqrt{15}}
$$
To remove the dependency in $x$ in the denominator, use that square roots are positive:
$$
|\sqrt{2x+7} - \sqrt{15}| = \frac {|2x-8|}{\sqrt{2x+7} + \sqrt{15}}
\le \frac {|2x-8|}{\sqrt{15}}
$$
</s>