LIMITS AND LIMITS AT INFINITY
Limit Review 1
Limits can be calculated 3 ways
• Numerically
• Graphically
• Analytically (direct substitution)
• Properties of limits (pages 56-58)
right theandleft both the fromnumber sameapproach must
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
2) Find
434lim 2
2
4
xxxx
x
432lim 2
5
xx
x
3) Find
732lim
7
xx
x
One sided limits
• left
• Right
• Limit existence
cxxfcx
)(lim
cxxfcx
)(lim
)(lim )(lim xfxfcxcx
4) Find xx
x 0lim
5) Find 2
2lim2
2
xxx
x
2222
2xxxx
x
Infinite Limits• Limit of f(x) increases or decreases without
bound as x approaches some number c
• Limit does not exist DNE• Vertical asymptotes occur when limits are
infinite• Limit as x approaches 1
or
6) Find
limx 1
?x 1
7) Find 1
2 lim1
x
xx
8) Find xx
1 xlim 2
0
9) Find 344
16 lim 2
2
21
xxxx
x
Limits at Infinity
• Identifies the behavior of the function as x tends to infinity
• If , then L is a horizontal asymptote
• Usually f(x) is rational• Limit as x approaches infinity
Lxfx
)(lim
10) Find 2
25 limxx
11) Find 32
6 lim xx
x
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
35 lim
2
xx
x
12 lim
2 x
xx xrecall 2x
14) Find and all asymptotes 4
82 lim 2
2
xxx
x
15) Find and all asymptotes 11 lim 2
2
xx
x
HOME WORKLimits Practice worksheet