Basic Limits:
lim x = b
x →b
lim c = c
x →b
1
1
1
1
lim = 0 lim = 0 lim− = −∞ lim+ = ∞
x → −∞ x
x →∞ x
x →0 x
x →0 x
x
x
lim+ ln x = −∞ lim ln x = ∞
lim e = 0 lim e = ∞
x → −∞
x →∞
x →∞
x →0
Evaluate limits of quotients using the following guidelines:
0
→0
non − zero
non − zero
→∞
0
or − ∞ or both (one from each side) depending on signs
∞
→ ∞ or − ∞ or both (one from each side) depending on signs
finite
finite
→0
∞
0
0
and
∞
are indeterminate forms done with algebra, trig or L’Hopital’s Rule
∞
Evaluate products using the following guidelines:
0 ⋅ finite → 0
and
non − zero ⋅ ∞ → ∞
0 ⋅∞ is an indeterminate form done by rewriting as a quotient and using L’Hopital’s Rule
Evaluate limits of exponential expressions using the following guidelines:
b∞ → ∞ and b−∞ → 0 for b > 1
b∞ → 0 and b−∞ → ∞ for 0 < b < 1
0+∞ → 0 and 0 −∞ =
1
→ ∞
0
or − ∞ or both (one from each side) depending on
signs
1∞ , 0 0 , and ∞0 are indeterminate forms whose logarithms have the form 0 ⋅∞ .
Finally ∞ − ∞ is an indeterminate form usually rewritten as a quotient to use either
algebra or L’Hopital’s Rule.