Section 11.1 Limits:
x2 − 9
. What is f (3)?
x−3
Can we approximate what f (x) is ‘close’ to x = 3?
1. Consider f (x) =
x
2.9
2.99
2.999
y
x
3.1
3.01
3.001
y
2. Limit: Let f be a function, and let a and L be real numbers. Assume that f (x) is defined for all
x near x = a.
Suppose that as x takes values very close (but not equal) to a (on both sides of a), the corresponding values of f (x) are very close (and possibly equal) to L.
And that the values of f (x) can be made as close as you want to L for all values of x that are
close enough to a.
Then the number L is the limit of the function f (x) as x approaches a which is written
lim f (x) = L.
x→a
3. Example: Let f (x) be defined as below, and answer the following two questions.
f (x) =
2x2 − 3x − 2
x−2
(a) What is f (2)?
(b) What is lim f (x).
x→2
1
4. Theorem Properties of Limits
Suppose
lim f (x) = L
x→a
and
lim g(x) = M .
x→a
Then,
1. lim cf (x) = c lim f (x) = cL (c ∈ R)
x→a
x→a
2. lim [f (x) ± g(x)] = lim f (x) ± lim g(x) = L ± M
x→a
x→a
x→a
3. lim [f (x)g(x)] = [lim f (x)][lim g(x)] = LM
x→a
x→a
lim f (x)
L
f (x)
x→a
=
=
4. lim
x→a g(x)
lim g(x)
M
x→a
(provided M 6= 0)
x→a
5. If p(x) is a polynomial, then lim p(x) = p(a)
x→a
6. lim [f (x)]r = [lim f (x)]r = Lr
x→a
x→a
(r ∈ R)
7. lim f (x) = lim g(x) if f (x) = g(x), for all x 6= a
x→a
x→a
5.
x3 + x2 − 2x
x→1
x−1
lim
6. Find lim g(x) where g(x) =
x→2
x3 − 2x2
.
x−2
2
7. Find
lim
x→0
2x − 1
x
8. Find lim f (x) and f (3) if
x→3
f (x) =
2x − 1 if x 6= 3
1
if x = 3
9. Find limx→1 f (x) and f (1) if
f (x) =
−x
√ if x ≤ 1
x if x > 1
3
10.
The amount of revenue generated (in billions of dollars) over a 5-year period from General
Electric can be approximated by the function f (x) = −4x2 + 20x + 150 where x = 0 corresponds to
the year 2005. find limx→5 f (x). And explain what this means!
11. Find lim [(x2 + 1) + (x3 − 1 + 3)]
x→2
12. Find lim (x3 + 4x)(2x2 − 3x)
x→−1
13. Suppose limx→2 f (x) = 3 and limx→2 g(x) = 4. Use the limit rules to find the following limits.
(a) lim [f (x) + 5g(x)]
x→2
(b) lim ln [f (x) + g(x)]2
x→2
4
14. Find
15. Find
x2 − x − 1
lim √
x→3
x+1
x2 − x − 12
x→−3
x+3
lim
16. Find
√
lim
x→1
17. Find
x−1
x−1
√
lim
x→4
x−2
.
x−4
5
18.
6