Calculus Cheat Sheet All

Calculus Cheat Sheet

Limits Definitions Precise Definition : We say lim  f ( x ) = L if  Limit at Infinity : We say lim  f ( x ) = L if we  x ®a

 x ®¥

> 0 there is a d  > 0 such that whenever 0 < x - a < d  then  f ( x ) - L < e  .

can make  f ( x ) as close to L to L as we want by

“Working” Definition : We say lim  f ( x ) = L

There is a similar definition for  lim  f ( x ) = L

if we can make  f ( x ) as close to L to L as we want

except we require x require x large and negative.

 by taking x taking x sufficiently close to a (on either side of a of a) without letting  x = a .

Infinite Limit : We say lim  f ( x ) = ¥ if we

for every

e 

 x ®a

Right hand limit : lim+  f ( x ) = L . This has  x ®a

the same definition as the limit except it requires  x > a . Left hand limit : lim-  f ( x ) = L . This has the  x ®a

taking x taking x large enough and positive.  x ®-¥ ®- ¥

 x ®a

can make  f ( x ) arbitrarily large (and positive)  by taking x taking x sufficiently close to a (on either side of a of a) without letting  x = a . There is a similar definition for  lim  f ( x ) = -¥  x ®a

except we make  f ( x ) arbitrarily large and

same definition as the limit except it requires negative. <  x a . Relationship between the limit and one-sided limits lim  f ( x ) = L Þ lim+  f ( x ) = lim- f ( x ) = L lim+  f ( x ) = lim- f ( x ) = L Þ lim  f ( x ) = L  x ®a

 x ® a

x ®a

 x ®a

 x ®a

x ®a

lim  f ( x ) ¹ lim- f ( x ) Þ lim  f ( x ) Does Not Exist

 x ®a +

x ®a

 x ®a

Properties Assume lim  f ( x ) and lim g ( x ) both exist and c is any number then,  x ®a

 x ®a

1. lim éëcf ( x ) ùû = c lim f ( x )  x ®a x ®a 2. lim éë f ( x ) ± g ( x ) ùû = lim f ( x ) ± lim g ( x )  x ®a

x ®a

x ®a

3. lim éë f ( x ) g ( x ) ùû = lim f ( x ) lim g ( x )  x ®a

x ®a

x ®a

 f ( x ) é  f ( x ) ù  xlim ®a 4. lim ê provided lim g ( x ) ¹ 0 ú=  x ®a  g x  x ®a lim g x ( ) ( ) ë û  x®a 5. lim éë f ( x ) ùû = élim f ( x ) ù  x ®a ë x ®a û n

n

6. lim é n  f ( x ) ù = n lim f ( x )  x ®a

ë

û

x ®a

Basic Limit Evaluations at ± ¥

 Note : sgn ( a ) = 1 if  a > 0 and sgn ( a ) = -1 if  a < 0 . 1. 2.

lim e x = ¥

 x ®¥

&

lim lim ln ( x ) = ¥

lim e x = 0

 x ®- ¥

&

 x ®¥

3. If  r  > 0 then lim

b

 x ®¥  x r 

lim+ ln ( x ) = - ¥

 x ®0

=0

4. If  r  > 0 and is real for negative x negative x b then li lim r  = 0 ®- ¥  x  x ®-¥ r 

5. n even : lim  x n = ¥  x ®± ¥

6. n odd : lim x n = ¥ & lim  x n = -¥  x ®¥

 x ®- ¥

7. n even : lim a x + L + b x + c = sgn ( a ) ¥ n

 x ®± ¥

8. n odd : lim a x n + L + b x + c = sgn ( a ) ¥  x ®¥

9. n odd : lim a x n + L + c x + d = - sgn ( a ) ¥

Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes.

 x ®-¥

© 2005 Paul Dawkins