Limit Properties

The limit properties (or limit laws) are simply the rules for finding the limit of a combination, like a sum or a product, from the limits of its pieces. All they say is that if you know where each piece is heading, you know where the combination is heading, which saves you from making a table every time. The formulas look like a wall of $\lim$ signs, but each one says the same thing for a different operation: $\lim[f \pm g]$ (add or subtract the limits), $\lim[f\,g]$ (multiply them), $\lim\frac fg$ (divide them, as long as the bottom isn't heading to $0$) and $\lim f^n$ (raise the limit to the power). It looks like a lot of rules, but it's really one idea repeated four times.

Think about why that should be true for a sum: if $f(x)$ is getting close to $3$ and $g(x)$ is getting close to $4$, then $f(x) + g(x)$ has nowhere to go but close to $7$. The same reasoning works for differences, products, quotients and powers, and that's all the limit laws for this lesson say.

Now start from the two simplest limits there are: $\lim_{x\to a} c = c$ (a constant stays put) and $\lim_{x\to a} x = a$ (if $x$ heads to $a$, then $x$ heads to $a$). Every polynomial is built from constants and $x$ using sums and products, so the rules let you build up its limit piece by piece. Here's what you get:

• Polynomials: $\lim_{x\to a} p(x) = p(a)$. Just substitute.
• Rational functions: $\lim_{x\to a}\frac{p(x)}{q(x)} = \frac{p(a)}{q(a)}$ when $q(a) \ne 0$.
• Roots: $\lim \sqrt[n]{f(x)} = \sqrt[n]{\lim f(x)}$ (for even $n$, the limit must be positive).

So for "nice" functions, the limit is just the value. That might seem like a letdown after the last lesson, where we made a big deal of the value at $a$ not mattering. But the interesting cases are exactly the ones where plugging in fails, usually because you get $\frac{0}{0}$. The quotient rule doesn't help there, since the denominator's limit is zero. Handling that is what the next lesson is about.

These rules are why small errors in a measurement give small errors in anything computed from it: if the inputs converge, the outputs converge.