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"Limits" 5 posts

#### The limit of rational expressions to infinity

another video on Limits to infinity:

#### Limits approaching infinity

An educational video With a limit of infinity:

#### Limit Rules

Here is a handout on limit rules:

Calculus I                            Carrie Diaz Eaton, Unity College

Special limits of sequences rules

1. When the base changes:

Discrete For an=n-k=1nk, k>0, thennan=0

Continuous or f(x) = x-k, k>0, then xf(x)=0

Example:

1. When the exponent changes:

For an=a0n, or f(t)=x0t

1. if -1<<1, then nan=0 or tf(t)=0

Example:

1. if >1, then nan DNE or tf(t) DNE

Example:

1. Rational expressions:

For an=bpnp+bp-1np-1+...+b1n+b0cqnq+cq-1nq-1+...+c1n+c0,

1. if p<q, then nan=0

Example:

1. if p=q, then nan=bpcq

Example:

1. if p>q and both p and q are positive or negative, then nan= (DNE)

Example:

1. if p>q and only one of p and q are positive, then nan=- (DNE)

Example:

1. Sandwich Principle

1. General case:

If  nan=ncn=L     and anbncn, then nbn=L

Example:

1. Special case, Alternating sequences with (-1)n:

If nan=0,     then nan=0,     otherwise the limit DNE

Example:

#### Properties of limits

Here is a handout on Limit properties:

Calculus I                                     Unity College

Properties of Limits

For real numbers, a (a can also be ), c, L, and M, given that taf(t)=L, tag(t)=M, then

 Property Example nac=c tacf(t)=c taf(t)=cL ta(f(t)+g(t))=taf(t)+tag(t)=L+M ta(f(t)-g(t))=taf(t)-tag(t)=L-M taf(t)g(t)=taf(t)tag(t)=LM taf(t)g(t)=LM,   as long as M0 For a natural number, m, tamf(t)=mL, as long asmf(t) and mL are defined for all t and ta(f(t))m=Lm, for any integer, m

WARNING!  CAREFUL OF INDETERMINATE FORMS, like - and , 00!

Note: -0     and     1!!!!!!

Special sequences (KNOW these!):

n(1/n) = 0   ,      n(e-n) = 0