AP Calculus AB · Summary

Unit 1 summary

1. A limit is about inputs near $a$, not at $a$. The two-sided limit exists only when both one-sided limits exist and agree.
2. Substitute first. A $\frac{0}{0}$ form means simplify (factor, rationalize, combine fractions) and substitute again; a nonzero number over 0 means the outputs are unbounded.
3. Squeeze: if $g \le f \le h$ near $a$ and $g$ and $h$ share the limit $L$ at $a$, then $f$ has limit $L$ at $a$. Also $\lim_{x \to 0}\frac{\sin x}{x} = 1$.
4. Continuity at $a$ needs three things: $f(a)$ is defined, the limit exists, and the two are equal. Check piecewise functions at their breakpoints.
5. IVT: if $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then $f(c) = N$ for some $c$ in $(a, b)$. Always name continuity.
6. A denominator zero that does not cancel gives a vertical asymptote; one that cancels gives a hole.
7. At infinity, compare dominant terms: equal degrees give the ratio of leading coefficients, and a lower degree on top gives 0.