The limit laws AB & BC

Suppose $\lim_{x\to a} f(x) = L$ and $\lim_{x\to a} g(x) = M$, both existing and finite. Then:

1. $\lim (f \pm g) = L \pm M$
2. $\lim (f\cdot g) = L\cdot M$
3. $\lim (c\cdot f) = c\cdot L$ for a constant $c$
4. $\lim \dfrac{f}{g} = \dfrac{L}{M}$, provided $M \ne 0$
5. $\lim \left(f\right)^n = L^n$, and $\lim \sqrt[n]{f} = \sqrt[n]{L}$ when the root is defined

The opening clause is not decoration. If either piece has no limit, the laws say nothing at all: not that the combination has no limit, simply that this tool does not apply. And $M = 0$ removes the quotient law entirely, which matters because that is exactly the case worth investigating.

Apply the laws repeatedly to a polynomial and every piece reduces to $\lim_{x\to a} x = a$ and $\lim_{x\to a} c = c$. The result is that for polynomials, and for rational functions wherever the denominator is nonzero,

$$\lim_{x\to a} f(x) = f(a).$$

So substitution is legitimate because the laws justify it, not because it is a habit. The same reasoning covers sines, cosines, exponentials, and logarithms on their domains: these are all continuous functions, and continuity is precisely the statement that substitution works.

The discipline is to check the result before reporting it. Substitution produces one of three outcomes: a determinate number, which is the answer; a $\frac{0}{0}$ form, which means more work; or a nonzero over zero, which signals unbounded behavior and an asymptote.

Sorting the three outcomes is the practical core of this topic:

1. A number. $\lim_{x\to 3}(x^2+1) = 10$. Done.
2. $\frac{0}{0}$. Indeterminate. Factor, rationalize, or simplify, then re-evaluate. Topic 1.6 is this case.
3. $\frac{k}{0}$ with $k \ne 0$. Not indeterminate. The magnitude grows without bound, so the limit does not exist and there is a vertical asymptote. Check each side for the sign.

Reporting "undefined" or "does not exist" for a $\frac{0}{0}$ is the characteristic error: it treats an unfinished calculation as a finished one. Reporting 0 for a $\frac{k}{0}$ is the mirror image, reading a zero denominator as though it made the fraction small.

Two failures of the hypotheses are worth naming.

First, splitting a limit that only exists as a whole. $\lim_{x\to 0}\frac{\sin x}{x} = 1$, but the numerator and denominator both go to 0, so the quotient law does not apply and $\frac{\lim \sin x}{\lim x}$ is $\frac{0}{0}$, not 1. The combined limit exists; the pieces do not cooperate.

Second, combining pieces that individually fail. $\frac{1}{x}$ has no limit at 0, and neither does $-\frac{1}{x}$, yet their sum is identically 0 and has limit 0. The laws do not run backwards: the whole can behave when the parts do not, which is why the hypotheses are stated on the parts.