LIMITS AND LIMITS AT
INFINITY
Limit Review 1
Limits can be calculated 3 ways
• Numerically 
• Graphically

 must approachsame number from both t heleft and theright


• Analytically (direct substitution)
• Properties of limits (pages 56-58)
Graphically
• Determine the value of y as x approaches
some number c
Formal Definition of a Limit
Analytic techniques
• Direct substitution
• Factoring- cancel common factors
• Rationalize- multiply by the conjugate
1) Find
lim 2 x 2  3 x  4
2) Find
x2  4x
lim 2
x 4 x  3 x  4
x 5
3) Find
x  2 3
lim
x 7
x7
One sided limits
• left
lim f ( x ) x  c
x c
• Right lim f ( x) x  c
x c 
• Limit existence
lim f ( x)  lim f ( x)
x c
x c
4) Find
lim
x 0
x
x
5) Find
x x2
lim
x 2
x2
2
x2
 x2
x2  
 x  2 x  2
Infinite Limits
• Limit of f(x) increases or decreases without
bound   or  as x approaches some
number c
• Limit does not exist DNE
• Vertical asymptotes occur when limits are
infinite
• Limit as x approaches 1
6) Find

?
lim
x 1
x 1
7) Find
2 x
lim
x 1
x 1
8) Find
1
lim x 
x 0
x
2
9) Find
6x  x 1
lim 
2
1
4
x
 4x  3
x 
2
2
Limits at Infinity
• Identifies the behavior of the function as x
tends to infinity
f ( x)  L , then L is a horizontal
• If lim
x 
asymptote
• Usually f(x) is rational
• Limit as x approaches infinity
10) Find
2
lim 5  2
x 
x
11) Find
6x
lim
x   2 x  3
Limits at Infinity Shortcuts
• If the degree is higher in the denominator,
then the limit is equal to zero
• If the degree is higher in the numerator then
the limit does not exist
• If the degrees are the same, then the limit
equals the ratio of the coefficients
BOBO BOTN EATS DC
12) Find
13) Find
5x2
lim
x   x  3
lim
x 
x
2x 1
2
recall
x2  x
2
x
 2 x  8 and all asymptotes
14) Find lim
x 
x2  4
2
x
 1 and all asymptotes
15) Find lim
x  x 2  1
HOME WORK
Limits Practice
worksheet