Summary Calculate the Dar

CALCULUS



AIRJEENEET

, Cheat Sheet Book

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 
for every   0 there is a   0 such that can make f  x  as close to L as we want by
whenever 0  x  a   then f  x   L   . taking x large enough and positive.

“Working” Definition : We say lim f  x   L There is a similar definition for lim f  x   L
x a x  

if we can make f  x  as close to L as we want except we require x large and negative.
by taking x sufficiently close to a (on either side
of a) without letting x  a . Infinite Limit : We say lim f  x    if we
xa

can make f  x  arbitrarily large (and positive)
Right hand limit : lim f  x   L . This has by taking x sufficiently close to a (on either side
x a
the same definition as the limit except it of a) without letting x  a .
requires x  a .
There is a similar definition for lim f  x   
x a
Left hand limit : lim f  x   L . This has the
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  lim f  x   lim f  x   L lim f  x   lim f  x   L  lim f  x   L
x a xa x a xa x a x a

lim f  x   lim f  x   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 cf  x   c lim f  x   f  x   lim f  x
xa xa 4. lim    x a provided lim g  x   0
 g  x   lim g  x
x a x a
x a
2. lim  f  x   g  x    lim f  x   lim g  x  n
5. lim  f  x    lim f  x  
n
xa xa xa
xa  xa 
3. lim  f  x  g  x    lim f  x  lim g  x  6. lim  n f  x    n lim f  x 
xa xa xa xa   xa



Basic Limit Evaluations at  
Note : sgn  a   1 if a  0 and sgn  a   1 if a  0 .
1. lim e x   & lim e x  0 5. n even : lim x n  
x x  x  

2. lim ln  x    & lim ln  x     6. n odd : lim x n   & lim x n  
x  x0  x  x  

3. If r  0 then lim
b
0 7. n even : lim a x    b x  c  sgn  a  
n
x  
x  x r
8. n odd : lim a x n    b x  c  sgn  a  
4. If r  0 and x r is real for negative x x 
b
then lim r  0 9. n odd : lim a x n    c x  d   sgn  a  
x  
x   x