**limits**

DEFINITION

examples

FIND THE MAXIMUM AREA OF A RECTANGLE

in a rectangle with a perimeter of 24,

2w + 2l = 24

l = 12 - w

A = (12 - w)w

= 12w - w²

Limits That Do Not Exist

left and right

unbound

oscillating

basic limits

**sally luken**

ap calculus

ap calculus

If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, then the limit of f(x) as x approaches c is L

https://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&docid=C94gY325gnzVeM&tbnid=fP5QdiFR30nTkM:&ved=0CAUQjRw&url=http%3A%2F%2Fen.wikipedia.org%2Fwiki%2F(%25CE%25B5%2C_%25CE%25B4)-definition_of_limit&ei=26AeUorRBZOw4AO5lIGIBw&bvm=bv.51495398,d.cWc&psig=AFQjCNGTHvAe-jkWycaAHOJLYDphlUvL2A&ust=1377825369309464

12 - w

W

the maximum value for A is when w = 6

"the limit of A as w approaches 6 is 36"

lim A = lim (12w - w²) = 36

w-->6 w-->6

http://www.google.com/url?sa=i&rct=j&q=the+limit+does+not+exist&source=images&cd=&docid=SAcgB1Sh6zfWlM&tbnid=jPy1Sx2Ri62uiM:&ved=0CAUQjRw&url=http%3A%2F%2Fwww.tumblr.com%2Ftagged%2Fthe-limit-does-not-exist&ei=SK4eUr_oDdKz4AOC5oHgCg&bvm=bv.51495398,d.cWc&psig=AFQjCNFOx0QBeu4P4m1ebNCZ72BXdrPN9A&ust=1377828753140342

EXAMPLES OF NONEXISTENT LIMITS

lim abs(x)/x

x-->0

lim sin (1/x)

x-->0

http://www.google.com/url?sa=i&rct=j&q=the%20limit%20of%20the%20(abs(x))%2Fx&source=images&cd=&docid=-DxXH4TQS_5hDM&tbnid=cqwBE7_yoSRsaM:&ved=0CAUQjRw&url=http%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F28268%2Fwhy-dont-these-limits-exist&ei=mL4eUuOpE7TJ4AO6iIGgBg&psig=AFQjCNEgCDQdtEVMMFrbFumoxRDVuzZhWg&ust=1377832956229899

no matter how close x gets to 0, there will always be both +/- x-values that yeild

f(x) = {1, -1}

SO LIMIT CANNOT EXIST

http://www.google.com/url?sa=i&rct=j&q=lim%20sin%201%2Fx%20as%20x%20approaches%200&source=images&cd=&docid=RhkLcEHtXv5TeM&tbnid=4pPvJ39WWrkpWM:&ved=0CAUQjRw&url=http%3A%2F%2Fwebpages.charter.net%2Fmwhitneyshhs%2Fcalculus%2Flimits%2Flimits.html&ei=isAeUriVDbTF4AORm4A4&psig=AFQjCNGrsdi2R8t57ilbQZtMmq3t8z4Mqg&ust=1377833449871581

no matter how close x is to 0, you could choose values of x that yeild

f(x) = {1, -1}

http://www.google.com/url?sa=i&source=images&cd=&docid=jllMcAsBqB9gNM&tbnid=BO-dXsuj7aO-sM:&ved=0CAgQjRwwADhR&url=http%3A%2F%2Fwww.cliffsnotes.com%2Fmath%2Fcalculus%2Fcalculus%2Flimits%2Finfinite-limits&ei=Jb0eUuH4AtS5sQT2zIC4Cw&psig=AFQjCNG3Jw_RAOv_JyGlUNOk5JuE_GMyLA&ust=1377832613112152

lim 1/x²

x-->0

because f(x) is not approaching a specific real # L as x approaches 0, you know that the limit does not exist

the existence or non-existence of f(x) at

x=c has no bearing on the existence of the

limit of f(x) as x--> c.

let b and c be real #s and n

be a positive integer:

1. lim b=b

2. lim x=c

3. lim x^n = c^n

4. lim √x = √c

x-->c

x-->c

x-->c

n

n

x-->c

estimating limits numerically

CONSIDER : lim(4x-2)

x-->2

from this information you can estimate that the limit is 6

so : f(x) = 4x-2

construct a table with

2 sets of values for x,

one from the left and

one right

the end

http://www.cengage.com/resource_uploads/downloads/1111427631_267947.pdf

http://www.mathsisfun.com/calculus/limits.html

http://archives.math.utk.edu/visual.calculus/1/limits.7/

http://www.rootmath.org/calculus/limits-that-fail-to-exist

and the assistance and approval of Nick Luken :)

as x approaches 2,

f(x) appears to approach 6

https://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&docid=2g2sf8Nz2M1szM&tbnid=DmONayTrIwGF4M:&ved=0CAUQjRw&url=http%3A%2F%2Fminimalshirt.spreadshirt.com%2Fkeep-it-simple-A7841883&ei=FgUgUqSfO5K24APrhIA4&bvm=bv.51495398,d.cWc&psig=AFQjCNEt71Vcg9Wer0go_f7Kt7ftvw1dhw&ust=1377916535277990