Understand limits and derivatives

Limit laws and their conditions

Check that the component limits exist before applying a limit law. For a quotient, check the limiting denominator before dividing.

Suppose finite limits $F=\lim_{x\to a}f(x)$ and $G=\lim_{x\to a}g(x)$ exist.

For constant k, the source’s four rules are $\lim(kf)=kF$, $\lim(f\pm g)=F\pm G$, $\lim(fg)=FG$, and $\lim(f/g)=F/G$ when $G\ne0$.

Its four table examples are $\lim_{x\to4}5x^3=320$, $\lim_{x\to3}(x^2-x)=6$, $\lim_{x\to2}(x\ln x)=2\ln2=\ln4$, and $\lim_{x\to-1}(x^2+1)/(2x+3)=2$.

Use continuity for direct substitution only where the expression is defined and continuous.

Forms such as $0/0$ and $\infty/\infty$ require rewriting; they do not determine a limit.