1. Calc 2.5 - Limits involving infinity
ex: lim x->0 1 x2
DNE
BUT… we can also say:
ex: lim x->0 1 x2
= 
We are NOT regarding  as a #
We are NOT saying the limit exists
We ARE expressing the particular way in which the limit doesn’t exist…

2. or: f(x)-->  as x-->a
General Notation
lim f(x) =  “The limit of f of x as x approaches a is infinity”… x-->a
or: f(x)-->  as x-->a
“f of x approaches infinity as x approaches a”
Negative  : lim = - 1 x2
x-->a a
x-->a-
x-->a+ a
One-Sided: lim f(x) = 
x-->a+

3. Vertical Asymptotes
“The line x = a is a vertical asymptote of the curve y = f(x) if y -->+ or - as x-->a(+/-)
ex: Find lim x-->3+ 2
x - 3
and lim x-->3-
Small +’s
Small -’s
x = 3
= 
= -

4. Limits AT Infinity… x2 - 1 ex: lim x--> x2 + 1 1.1 -1 25
-1.1

5. OR lim f(x) = L x-->-
Horizontal Asymptote
“The line y = L is a horizontal asymptote of the curve y = f(x) if either:
lim f(x) = L x-->
OR lim f(x) = L x-->-
lim f(x) = x-->-1-
lim f(x) = x-->-1+ -1 2 1
lim f(x) = - x-->2-
lim f(x) =  x-->2+
lim f(x) = 0 x-->-
lim f(x) = 2 x-->

6. 1 1
ex: Find lim x-->
and lim x-->- x x
General Law: “If n is a + integer then:
lim x--> 1 xn
= 0
AND lim x-->-
Technique: infinity of a rational function:
Divide BOTH numerator AND denominator by the highest power of x in the DENOMINATOR…
ex: lim x-->
3x2 - x - 2
5x2 + 4x + 1

7. [3x2 - x - 2] / x2 =
3 - 1/x - 2/x2
5 - 4/x + 1/x2
lim x-->
ex: lim x-->
Goes to 0
[5x2 + 4x + 1] / x2 = 3 5
x x
ex: lim x x x--> =
lim (x2 + 1) - x2 x-->
x x =
lim x-->
x x
/ x =
1/x
1 + 1/x2 + 1
lim x--> = 2

8. General Rule: lim ax = 0 for any a > 1 x -∞
lim 1/x = -∞ x0-
= e-∞ as x 0-
Evaluate lim e1/x x0- = 1
e ∞ =
General Rule: lim ax = 0 for any a > x -∞
Infinite Infinity
General notation: lim f(x) = ∞ x∞
“as x gets big, f(x) gets big”
Other Examples:
lim f(x) = -∞ x∞
lim f(x) = ∞ x-∞
lim f(x) = -∞ x-∞
“as x gets big, f(x) gets small”
“as x gets small, f(x) gets big”
“as x gets small, f(x) gets small”

9. HW 2.5 – pg 140 - #’s 1-3 all, 5, 13, 17-20 all, 31-33 all.
ex:
lim x3 = ∞ x∞
lim x3 = -∞ x-∞
lim ex = ∞ x∞
≠ (∞ - ∞ )
ex: Find lim (x2 – x) x∞
= lim x (x – 1) x∞
= ∞ (∞ - 1)
= ∞ = ∞ -1
ex: Find lim x∞
x2 + x
3 - x
/ x =
x + 1
3/x - 1
lim
x∞ =
- ∞
HW 2.5 – pg #’s 1-3 all, 5, 13, all, all.

10. 
Calc 2.5 !