Substitution of a absolute value with variable K, when proving a limit, how is the process called? I am trying to understand for quite some time now, the concept of substitution with K when proving the limit. Is it to find delta by changing the upper bound restriction for epsilon? What is the name of this kind of substitution?
The sample is Example 3 in http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx 
 A: Yes, it is to find a value for $\delta$. No, there's no special name for this method. But there are some useful ideas /methods / approaches (whatever we wanna call them) at play here.
One idea is that we are going to try to find a value of $\delta$ that is proportional to $\varepsilon$, i.e. in the form $\displaystyle \delta=\frac{\varepsilon}{K}$. Nobody says that $\delta$ has to be of this form! But it's a nice form, so let's give it a try — and if we succeed, then all is good. What also helps here is that the definition of limits only requires that for each $\varepsilon$ there exists an appropriate $\delta$, but it's not unique — in fact, it can't be unique because if we find any one value for $\delta$, any smaller positive value will work too. So we're "solving" for $\delta$ in a somewhat loose sense — not for the only answer, but for anything that works. That's what gives us the flexibility to (try to) choose a convenient form for it.
And another idea is, as you pointed out, of finding an upper bound, although not for $\varepsilon$, but some other functions involved. More specifically, since we're going to consider only a (relatively small) interval $(x-\delta,x+\delta)$ on the $x$-axis, the function $|x+5|$ is bounded on it, so we can take $K$ to be an upper bound for $|x+5|$ on this interval (technically speaking, on the closed interval $[x-\delta,x+\delta]$ or some larger closed interval containing it). Then $|x+5|<K$ within $(x-\delta,x+\delta)$ and the rest of the proof follows.
