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  • Created 14 Jun 2016

"Limits" 5 posts Sort by created date Sort by defined ordering View as a grid View as a list

The limit of rational expressions to infinity

another video on Limits to infinity:

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Timothy John Beaulieu onto Limits

Limits approaching infinity

An educational video With a limit of infinity: 

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Timothy John Beaulieu onto Limits

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:

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Profile picture of Timothy John Beaulieu

Timothy John Beaulieu onto Limits

Video on limits

Here is a educational video on limits:

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Profile picture of Timothy John Beaulieu

Timothy John Beaulieu onto Limits

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

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Timothy John Beaulieu onto Limits