Section 2.3 · Limit Laws

Overview

Limit laws let you break a messy limit into simple pieces: add, subtract, multiply, divide, pull out constants, raise to powers, and take roots.

Goal for this section: Be able to combine known limits, and to turn a \(0/0\) into a form where substitution works by factoring or rationalizing.

The Limit Laws

  • Sum: \(\lim (f + g)=\lim f + \lim g\)
  • Difference: \(\lim (f- g)=\lim f - \lim g\)
  • Constant multiple: \(\lim (c\,f)=c\cdot\lim f\)
  • Product: \(\lim (fg)=(\lim f)(\lim g)\)
  • Quotient: \(\lim (f/g)=(\lim f)/(\lim g)\) if \(\lim g\neq0\)
  • Powers/Roots: \(\lim (f^n)=(\lim f)^n\), \(\lim \sqrt[n]{f}=\sqrt[n]{\lim f}\) when defined
Example: Polynomial

\(\displaystyle \lim_{x\to 2}\big(3x^2-5x+1\big)=3(2)^2-5(2)+1=3\)

Example: Rational with nonzero denominator

\(\displaystyle \lim_{x\to 4}\frac{x^2-1}{x+3}=\frac{\lim_{x\to 4} x^2-1}{\lim_{x\to 4} x+3}=\frac{16-1}{7}=\frac{15}{7}\)

Fixing \(0/0\)

Example: Factor & cancel
\(\displaystyle \lim_{x\to 3}\frac{x^2-9}{x-3}\).

Factor: \(\frac{(x-3)(x+3)}{x-3}=x+3\) (for \(x\ne3\)). Now substitute: \(3+3=6\).

Example: Rationalizing a root
\(\displaystyle \lim_{x\to 0}\frac{\sqrt{4+x}-2}{x}\).

Multiply top/bottom by \(\sqrt{4+x}+2\): \(\frac{(4+x)-4}{x(\sqrt{4+x}+2)}=\frac{1}{\sqrt{4+x}+2}\to \frac14\).

Interactive: Why “factor & cancel” works for limits
original \(f(x)\)
simplified \(g(x)\)
Left probe: \(x\to a^-\), \(y\approx\) —    Right probe: \(x\to a^+\), \(y\approx\) —    \(g(a)=\) —

Conclusion: Near \(a\) (but not at \(a\)), the functions are equal. Canceling common factors only changes the value at the hole, not the limit.

Combining limits of \(f\) and \(g\)

We can use information about \(f\) and \(g\) and to compute limits of expressions like \(f(x)+g(x)\), \(f(x)g(x)\), or \(\frac{f(x)}{g(x)}\) at a given \(a\).

Example

Suppose from the graphs you find \( \lim_{x\to 2} f(x)=3\) and \( \lim_{x\to 2} g(x)=-\tfrac12\).

  • \(\displaystyle \lim_{x\to2}\big(4f(x)-3g(x)\big)=4(3)-3(-\tfrac12)=12+\tfrac32=\tfrac{27}{2}\).
  • \(\displaystyle \lim_{x\to2}\big(f(x)g(x)\big)=3\cdot(-\tfrac12)=-\tfrac32\).
  • \(\displaystyle \lim_{x\to2}\frac{f(x)}{g(x)}=\frac{3}{-1/2}=-6\). This is allowed since \(\lim_{x\to 2} g(x) \ne0\).
Example: Wacky Limits