Why not teach $\lim\limits_{x \to 0} \frac { \sin({ K }_{ 1 }x) }{ { K }_{ 2 }x } =\frac { { K }_{ 1 } }{ { K }_{ 2 } } $? I was doing some Calculus homework the other day when I had a realization. As we know, the limit of sin(x) divided by x as x approaches 0 is 1:
$\underset { x\to 0 }{ lim } \frac { sin(x) }{ x } =1$
However, sometimes problems are presented to us that are similar but not exactly the same, such as this:
$\underset { x\to 0 }{ lim } \frac { sin(2x) }{ 3x } =??$
As I was doing Calculus problems, and I was repeatedly given situations like the above one, I began to notice a pattern:
$\underset { x\to 0 }{ lim } \frac { sin({ K }_{ 1 }x) }{ { K }_{ 2 }x } =\frac { { K }_{ 1 } }{ { K }_{ 2 } } $
I tested this numerous times with many different values, and this pattern always seemed to hold true. I then searched through my textbook (and online articles) to see if this was a standard identity that I had missed, but I couldn't find mention of it anywhere. Earlier today I asked my Calculus teacher about this pattern, and he also said that he'd never seen this before. It's hard for me to believe that this relationship/identity isn't already known (I'm just a high school student, after all. Some professor should have already found this!), so why isn't this taught more? Instead of being taught to memorize $\underset { x\to 0 }{ lim } \frac { sin(x) }{ x } =1$, why aren't we taught to memorize $\underset { x\to 0 }{ lim } \frac { sin({ K }_{ 1 }x) }{ { K }_{ 2 }x } =\frac { { K }_{ 1 } }{ { K }_{ 2 } } $? Is there some edge case or odd scenario where this identity doesn't work?

TLDR; 
Is there a mathematical reason (for example, edge-cases where this doesn't work) why Calculus students, when they're learning about limits, are taught $\underset { x\to 0 }{ lim } \frac { sin(x) }{ x } =1$ instead of $\underset { x\to 0 }{ lim } \frac { sin({ K }_{ 1 }x) }{ { K }_{ 2 }x } =\frac { { K }_{ 1 } }{ { K }_{ 2 } } $?

Note:
The answer to this question may be obvious, but I have done some research online and have not been able to find an answer. I'm just a high school student trying to figure out if there's a reason why this isn't taught (for example, is there an edge case where this doesn't work). Please let me know if more information is needed. Thanks in advance!
 A: I think it should be fairly easy to see that
$$
\lim_{x\to 0}\frac{\sin(K_1x)}{K_1x}=\lim_{y\to 0}\frac{\sin(y)}{y}=1
$$
setting $y=K_1 x$ and noting that $y\to 0$ as $x\to 0$.  Then changing the bottom $K_1$ into a $K_2$ is nothing more than a constant multiplication by $K_1/K_2$, which gives your result.  
As for why it's not taught, the prevailing taste in (many parts of) mathematics is to present results in the most economic fashion possible.  Proving that $$\lim_{x\to 0}\frac{\sin(x)}{x}=1$$ requires some non-trivial arguments, whereas getting from there to $$\lim_{x\to 0}\frac{\sin(K_1x)}{K_2x}=\frac{K_1}{K_2}$$ is nothing more than some standard arguments using limits.  So we present the 'hard' result, and leave the 'easy' corollary for the student to derive on the fly.  
From a pedagogical point of view, the fact that you haven't been taught this result directly means that you've had some practice manipulating limits and have discovered it for yourself, which is far more valuable than being taught it by rote.  
A: This is an important property of limits:
$\lim_\limits{x\to a} \frac {kf(x)}{g(x)} =k \lim_\limits{x\to a} \frac {f(x)}{ g(x)}$
This is another one:
$\lim_\limits{x\to 0} \frac {f(kx)}{g(kx)} =\lim_\limits{x\to 0} \frac {f(x)}{ g(x)}$
Your question is a specific example of the two of these more fundamental properties acting together:
$\lim_\limits{x\to 0} \frac {\sin(K_1x)}{K_2 x} = \frac {K_1}{K_2}\lim_\limits{x\to 0} \frac {\sin(K_1x)}{K_1 x} = \frac {K_1}{K_2}\lim_\limits{x\to 0} \frac {\sin(x)}{ x}$
A: Memorizing formulas vs Understanding the logic
Especially these days students are looking for formulas or clever mnemonics to solve problems. You do not need to memorize 
$$
\lim \limits_{x \rightarrow 0} \frac{\sin (K_1 x)}{K_2x} = \frac{K_1}{K_2}
$$ 
when you know 
$$
\lim \limits_{x \rightarrow 0} \frac{\sin x}{x} = 1
$$ 
Instead of memorizing the first one, as Donkey 2009 explained, you can see it as a result of the second one.
Slightly related note
Recently, I saw an engineering student who uses a mnemonics to remember what $\sin 30^\circ$ is. I said you can easily derive it by drawing an equilateral triangle and one of the angle bisectors of it. He was surprised seeing this derivation. Instead of memorizing all $\sin 30, \sin 60$ etc, you can actually understand where they come from.

