1. Prep Book Chapter 3 - Limits of Functions
Ex: Investigate the behavior of function f defined by: f(x) = (x2 - x + 2) for values of x near 2…
The closer to 2 x is, the closer to 4 f(x) is. So we write:
And say: “The limit of f(x) as x approaches a = L”
General notation:

2. The limit of f(x) as x approaches a = L IF we can get f(x) as close to L as we want by taking x as close to a as we want, BUT NOT EQUAL TO a (x  a)
Alternate notation for :
ex: Guess the value of: 1 x L .5

3. One-Sided Limits a a “Left-Hand Limit” “Right-Hand Limit” x x x x x xor
“As x approaches a from the left…”
“As x approaches a from the right…” a a
“Left-Hand Limit”
“Right-Hand Limit” x x x x x x

4. The left and right hand limits must be equal for the overall (2-sided) limit to exist…

5. ex: Find if it exists.
Does Not exist (DNE)

6. IF c is constant and lim f(x) and lim g(x) exist then…
1. lim [f(x) + g(x)] = lim f(x) + lim g(x)
x->a
“The limit of a sum (or diff) is the sum (or diff) of the limits”
“The limit of a constant times a function is the constant times the limit of the function.”
2. lim [cf(x)] = c lim f(x)
x->a
x->a
3. lim [f(x)g(x)] = lim f(x) • lim g(x)
“The limit of a product is the product of the limits”
(= 0)
lim f(x)
lim g(x)
x->a
f(x)
g(x)
4. lim =
“The limit of a quotient is the quotient of the limits”

7. ex: Use the limit laws and the graphs of f and g below to evaluate the following limits if they exist…
a) lim [f(x) + 5g(x)]
x->-2 f g
b) lim [f(x)g(x)]
x->1
c) lim
x->2
f(x)
g(x)

8. Additional Laws lim [f(x)] = lim f(x) n 5. Power Law:
x->a n
5. Power Law:
6. Special Limits:
lim x = a
x->a n
lim x = a
x->a n
lim c = c
x->a
lim x = a
x->a
7. Root Law:
lim f(x) = lim f(x)
x->a n
(where n is a positive integer)

9. “If f is a polynomial or rational function and a is in the domain of
f then”:…
lim f(x) = f(a)
x->a
Direct Substitution
ex: lim 2x2 - 3x + 4 = [2(5)2 - 3(5) + 4] = 39
x->5
“Continuous at a”
(x + 1)(x - 1)
x - 1
lim
x->1
lim
x->1
x2 - 1
x - 1 =
lim (x + 1) = 2
x->1 =

10. (3 + h)2 = (3 + h)(3 + h) = [9 + 6h + h2]
ex: Evaluate lim
h-->0
(3 + h)2 - 9 h
- 9
(3 + h)2 = (3 + h)(3 + h) = [9 + 6h + h2]
6h + h2 h =
lim
h-->0 =
h (6 + h) h
lim
h-->0 =
lim
h-->0
(6 + h)
= 6
t2 + 9
ex: Find lim
t-->0
- 3 t2
t2 + 9
+ 3 =
(t2 + 9) - 9 t2
t2 + 9
+ 3
t-->0
lim = t2
t-->0
lim
t2 + 9
+ 3 =
t-->0
lim 1
t2 + 9
+ 3 1 =
t-->0
lim
9 + 3 6

11. If f(x) < g(x) when x is near a (except possibly at a), and the limits of f and g both exist as x approaches a, then:
lim f(x) < lim g(x)
x-->a
x-->a
The Squeeze Theorem
If f(x) < g(x) < h(x) when x is near a (except possibly at a) and:
lim f(x) = lim h(x) = L
x-->a
x-->a
then…
lim g(x) = L
x-->a

13. 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… 13

14. 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+ 14

15. 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
= 
= - 15

16. Limits AT Infinity… x2 - 1 ex: lim x--> x2 + 1 1.1 -1 25
-1.1 16

17. 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-->

18. 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

19. [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

20. 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”

21. ∞ 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∞ =
- ∞

22. Trig Identities and Special Limits
Additional Rules:
Pythagorean Identity: