Transcript Limits and Limits at infinity
L IMITS AND L IMITS AT INFINITY
Limit Review 1
Limits can be calculated 3 ways • • Numerically must approach same number from both the left and the right Graphically • 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
x
5 2
x
2 3
x
4 2) Find
x
lim 4
x
2
x
2 4
x
3
x
4
3) Find
x
lim 7
x x
2 7 3
One sided limits • left
x
lim
c
f
(
x
)
x
c
• Right
x
lim
c
f
(
x
)
x
c
• Limit existence
x
lim
c
f
(
x
)
x
lim
c
f
(
x
)
4) Find
x
lim 0
x x
5) Find
x
lim 2
x
2
x
x
2 2
x
2
x x
2 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
x
lim 1 2
x
x
1
8) Find
x
lim 0 x 2 1
x
9) Find lim
x
1 2 4 6
x x
2 2 4
x x
1 3
Limits at Infinity • • Identifies the behavior of the function as x tends to infinity If , then
x
lim asymptote L is a horizontal • • Usually f(x) is rational Limit as x approaches infinity
10) Find
x
lim 5 2
x
2
11) Find
x
lim 2
x
6
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
x
lim
x
5
x
2 3 13) Find
x
lim
x
2
x
2 1
recall
x 2
x
14) Find
x
lim
x
2
x
2 2
x
4 8 and all asymptotes
15) Find
x
2
x
lim
x
2 1 1 and all asymptotes
H OME W ORK
Limits Practice worksheet