Why do these laws work?
By definition of limit, if $\displaystyle\lim_{x \to a} f(x) = L$ and $\displaystyle\lim_{x \to a} g(x) = M$, then $f(x)$ is close to $L$ and $g(x)$ is close to $M$ whenever $x$ is close to $a$.

Adding two numbers close to $L$ and $M$ gives a number close
to $L+M$.

Likewise, taking the difference of two numbers close to $L$ and $M$
gives a number close to $LM$.

Taking the product gives a number close to $L\cdot M$, and taking the ratio
gives a number close to $L/M$ as long as $M \ne 0$.

Multiplying $f(x)$ by a constant $c$ gives a number close to
$cL$. This gives the rest of the limit laws.
When $M=0$ and/or $L=0$
The limit of a quotient rule with $M=0$ is more subtle. If $f(x)$ is close to 4
and $g(x)$ is close to 0, what can you say about $f(x)/g(x)$? It might
be $3.99/(.01) = 399$, or it might by $4.01/(.01) = 401$. Dividing
by a number close to zero might give you a big positive ratio, or a
big negative ratio. Without knowing more about $g(x)$, you just can't
tell.
Things get even wilder when both $f(x)$ and $g(x)$ are both
close to zero. In that case, $$\frac{f(x)}{g(x)} = \frac{\hbox{
small number}}{\hbox{small number}}.$$
This ratio could be
$$ \frac{.000001}{.001} = 0.001,$$
or it could be
$$\frac{.001}{.000001} = 1000, $$
or it could be
$$\frac{.001}{0.000001} = 1000.$$
Without understanding the numerator and denominator better, we just can't tell how
these "0/0" limits will behave. We'll learn more about these kinds of limits (called "indeterminate forms") in a later module. For now, we'll just file them under "do more work".
