4. Computation Rules and Standard Limits

Computation Rules

Theorem (Sum Rule)

Suppose

${\lim }_{x\to \infty }f(x)=a$ and ${\lim }_{x\to \infty }g(x)=b$

then:

${\lim }_{x\to \infty }f(x)+g(x)={\lim }_{x\to \infty }f(x)+{\lim }_{x\to \infty }g(x)=a+b$

Theorem (Scale Rule)

${\lim }_{x\to \infty }cf(x)=c{\lim }_{x\to \infty }f(x)=ca$

Theorem (Product Rule)

${\lim }_{x\to \infty }f(x)g(x)=({\lim }_{x\to \infty }f(x))({\lim }_{x\to \infty }g(x))=ab$

Theorem (Quotient Rule)

${\lim }_{x\to \infty }\frac{f(x)}{g(x)}=\frac{{\lim }_{x\to \infty }f(x)}{{\lim }_{x\to \infty }g(x)}=\frac{a}{b}$

if $b\ne 0$

Theorem (Composition Rule)

Suppose

${\lim }_{x\to \infty }f(x)=a$

and $h(x)$ does not have a jump at $x=a$, then:

${\lim }_{x\to \infty }h(f(x))=h({\lim }_{x\to \infty }f(x))=h(a)$

Standard Limits

Theorem

Suppose $r>0$ and $k$ is positive integer

${\lim }_{x\to \pm \infty }c=c$

${\lim }_{x\to \infty }\frac{1}{x^r}=0$

${\lim }_{x\to -\infty }\frac{1}{x^k}=0$

${\lim }_{x\to \infty }{e}^{-rx}=0$

${\lim }_{x\to -\infty }{e}^{rx}=0$