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MATH 1225 Lecture Notes - Lecture 10: Asymptote, Rational Number

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28 May 2018

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Sec. 2.6 Limits at Infinity; Horizontal Asymptotes

EX: Finding horizontal asymptotes from a graph

1.

 

x

x

xf sin4



2.

 

2

1

2 x

xf

3.

 

xxf 1

tan



• Intuitive Definition of a Limit at Infinity: Let

f

be a function defined on some interval

 

,a

. Then

 

Lxf

x



lim

means that the values of

 

xf

can be made arbitrarily close to

L

by requiring

x

to be

sufficiently large.

• Intuitive Definition of a Limit at Negative Infinity: Let

f

be a function defined on some interval

 

a,

.

Then

 

Lxf

x



lim

means that the values of

 

xf

can be made arbitrarily close to

L

by requiring

x

to be

sufficiently large negative.

• Definition of a horizontal asymptote: The line

Ly 

is called a horizontal asymptote of the curve

 

xfy 

if either

 

Lxf

x



lim

or

 

Lxf

x



lim

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EX: Sketch the graph of

 

xf

and evaluate the limits

1.

     

xfxf

x

xf xx 

lim,lim,

1

2.

     

xfxfexf xx

x



lim,lim,

3.

     

xfxfexf xx

x





lim,lim,

4.

     

xfxfxxf xx 



lim,lim,tan 1

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Document Summary

Ex: finding horizontal asymptotes from a graph xf sin4 x x xf. Intuitive definition of a limit at infinity: let f be a function defined on some interval lim x means that the values of can be made arbitrarily close to l by requiring x to be. Intuitive definition of a limit at negative infinity: let f be a function defined on some interval . A, can be made arbitrarily close to l by requiring x to be means that the values of. Xf sufficiently large negative: definition of a horizontal asymptote: the line if either lim x. Ly is called a horizontal asymptote of the curve y . Xf lim x xf x e lim, x xf. Xf lim x xf e x lim, x xf. Xf lim x xf tan 1 x lim, x xf. If is a rational number, then lim x.

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ISBN: 9781319050733

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ISBN: 9781337687805

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ISBN: 9781319050740

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