The Limit Laws
Once you know what a limit
is, you almost never need a table again. Limits obey a handful of tidy
laws that let you break a big limit into smaller pieces. Suppose both
of these limits exist:
\lim_{x \to c} f(x) = L, \qquad \lim_{x \to c} g(x) = M.
Then the limit of a combination is the same combination of the limits.
The core laws
-
Sum / difference:
\lim (f \pm g) = L \pm M — the limit passes through a
+ or -.
-
Constant multiple:
\lim (k\,f) = k\,L — pull constants straight out.
-
Product: \lim (f \cdot g) = L \cdot M.
-
Quotient:
\lim \dfrac{f}{g} = \dfrac{L}{M}, provided
M \ne 0.
-
Power: \lim f^{\,n} = L^{\,n}.
Two everyday building blocks anchor them all:
\lim_{x \to c} k = k (a constant stays put) and
\lim_{x \to c} x = c (the identity heads where
x heads).
The sum law, in pictures
Why does \lim(f + g) = L + M? Because if
f(x) is hugging L and
g(x) is hugging M as
x \to c, their sum can only hug
L + M. Slide x toward
c = 1 below and watch the three heights stack up.
A worked combination
These laws chain together. Suppose
\lim_{x\to c} f = 4 and
\lim_{x\to c} g = 3. Then:
\lim_{x\to c}\bigl(2f + g^2\bigr) = 2\lim f + \bigl(\lim g\bigr)^2 = 2(4) + 3^2 = 17.
Constant multiple, then sum, then power — each law applied in turn. The reference
curves below show two functions and their sum so you can see the heights really do add.
Watch Sal walk through them