Additional Notes on Limits
1. If the left-hand limit of a function is not equal to the right-hand limit of the function, then the
limit does not exist.
2. A limit equal to infinity is not the same as a limit that does not exist, but sometimes you will see
the expression “no limit,” which serves both purposes. If lim f  x    , the limit, technically,
xa
does not exist.
k
k
k
  , lim   , and lim does not exist.
x 0 x
x 0 x
x 0 x
k
k
k
4. If k is a positive constant, then lim 2   , lim 2   , and lim 2   .
x 0 x
x 0 x
x 0 x
k
k
5. If k and n are positive constants, x  1 , and n  0 , then lim n  0 and lim n  0 .
x  x
x  x
6. If the highest power of x in a rational expression is in the numerator, then the limit as x
3. If k is a positive constant, then lim
approaches infinity is infinity.
7. If the highest power of x in a rational expression is in the denominator, then the limit as x
approaches infinity is zero.
8. If the highest power of x in a rational expression is in the same in both the numerator and
denominator, then the limit as x approaches infinity is the coefficient of the highest term in the
numerator divided by the coefficient of the highest term in the denominator.
9.
lim
x 0
sin x
1
x
10. lim
cos x  1
0
x
11. lim
sin ax
a
x
12. lim
sin ax a

sin bx b
x 0
x 0
x 0
( x is in radians, not degrees)


2.
lim
1
x2
4.
lim
1
x
lim
1
x  x
6.
lim
lim
8x2  4 x  1
x  16 x 2  7 x  2
8.
lim
lim
5 x 7  3x
x  16 x 6  3 x 2
10.
lim
11.
5 x7  3x
lim
x  16 x 7  3 x 2
12.
lim
13.
lim
sin 5 x
x  0 sin 4 x
14.
lim
15.
4x
lim
x 0 tan x
16.
5  h
lim
1.
lim  x 2  1
3.
lim
5.
7.
9.
x 5
x0
x  1 
1
x
x 0
x 
3x  5
x  7 x  2
 x10  70 x5  x3
x  33 x10  200 x8  1000 x 4
5 x 6  3x
x  16 x 7  3 x 2
x 0
sin 3 x
x
x2
x 0 1  cos 2 x
h 0
2
h
 25