Delta, Epsilon Definition of a Limit

\$\forall\epsilon\gt0\ \ \exists\delta\gt\0\ \ s.t. \forall\epsilon\mathbb{R} \ with\ 0 \lt\ |x - a|\ \lt \ \delta,\ we\ have\ |f(x) - L|\ \lt\ \epsilon\$
Epsilon Delta Definition of a Limit

Intermediate Value Theorem

Suppose \$f\$ is continuous on \$[a,b\$], and \$N\$, a number b/t \$f(a)\$ and \$f(b)\$, where \$f(a)\ \ne\ f(b)\$, then there exists a number \$c\$ in \$(a,b)\$ s.t \$f(c) = N\$.

Intermediate Value Theorem

Solving for Vertical Asymptotes

\$f(x) = frac{x+4}{(x+4)(x-8)}\$

  • -4 is removable

  • 8 is non-removable

In this function there would be a hole at -4

Limits to Memorize

  • \$\lim_{x \to 0^+}1/x = \infty\$

  • \$\lim_{x \to 0^-}1/x = -\infty\$

  • \$\lim_{x \to \infty}1/x = 0\$

  • \$\lim_{x \to 0}\sinx/x = \lim_{x \to 0}x/\sinx = 1\$

  • \$\lim_{x \to 0}\frac{cosx - 1}{x} = 0\$

  • \$\lim_{x \to \infty}(1 + 1/x)^x = e \approx 2.7.8\$