Limits Chapter 1 & 2
Note that Remember that the symbol Σ (sigma) represents “the sum of.”
Limits The word “limit” is used in everyday conversation to describe the ultimate behavior of something, as in the “limit of one’s endurance” or the “limit of one’s patience.”
In mathematics, the word “limit” has a similar but more precise meaning.
Limits Given a function f(x), if x approaching 3 causes the function to take values approaching (or equalling) some particular number, such as 10, then we will call 10 the limit of the function and write
In practice, the two simplest ways we can approach 3 are from the left or from the right.
Limits For example, the numbers 2.9, 2.99, 2.999, ... approach 3 from the left, which we denote by x→3 –, and the numbers 3.1, 3.01, 3.001, ... approach 3 from the right, denoted by x→3 +. Such limits are called one-sided limits.
Example– FINDING A LIMIT BY TABLES Use tables to find Solution :
We make two tables, as shown below, one with x approaching 3 from the left, and the other with x approaching 3 from the right.
Limits IMPORTANT! This table shows what f (x) is doing as x approaches 3. Or we have the limit of the function as x approaches We write this procedure with the following notation.
lim 2x 4 10 x3
10
lim f (x) L
Def: We write
3
x c
or as x → c, then f (x) → L if the functional value of f (x) is close to the single real number L whenever x is close to, but not equal to, c. (on either side of c). x
2
2.9
2.99
f (x)
8
9.8
9.98
7
2.99 9 9.99 8
3 ?
3.00 1
10.002
3.01
3.1
4
10.02
10.2
12
Limits As you have just seen the good news is that many limits can be evaluated by direct substitution.
Limit Properties These rules, which may be proved from the definition of limit, can be summarized as follows. For functions composed of addition, subtraction, multiplication, division, powers, root, limits may be evaluated by direct substitution, provided that the resulting expression is defined.
lim xď‚Žc
9
f (x)  f (c)
Examples – FINDING LIMITS BY DIRECT SUBSTITUTION 1.
lim
x
4 2
Substitute 4 for x.
x4
2.
lim x6
62 36 x2 4 x3 63 9
Substitute 6 for x.
Direct Substitution
Direct Substitution
Direct Substitution But be careful when a quotient is involved. x2 x 6 0 lim x2 x2 0 Graph it.
Which is undefined! But the limit exist!!!! What happens at x = 2?
x2 x 6 (x 3)(x 2) lim lim lim (x 3) 5 x2 x2 x2 x2 x2 x2 x 6 NOTE : f ( x ) graphs as a straight line. x2 13
One-Sided Limit We have introduced the idea of one-sided limits. We write
lim f ( x) K
x c
and call K the limit from the left (or lefthand limit) if f (x) is close to K whenever x is close to c, but to the left of c on the real number line. 14
5
One-Sided Limit We write
lim f ( x) L
x c
and call L the limit from the right (or righthand limit) if f (x) is close to L whenever x is close to c, but to the right of c on the real number line.
The Limit Thus we have a left-sided limit:
And a right-sided limit:
lim f ( x) K
x c
lim f ( x) L
x c
And in order for a limit to exist, the limit from the left and the limit from the right must exist and be equal.
Example f (x) = |x|/x at x = 0
lim
x 1 x
lim
x 1 x
x0
x0
0
The left and right limits are different, therefore there is no limit. 17
Infinite Limits Sometimes as x approaches c, f (x) approaches infinity or negative infinity. Consider
lim
x2
1
x 2
2
From the graph to the right you can see that the limit is ∞. To say that a limit exist means that the limit is a real number, and since ∞ and - ∞ are not real numbers means that the limit does not exist. 18
Indeterminate Forms ∞/∞, -∞/ ∞, 0/0
Dealing with Indeterminate Forms Factor and Reduce
T# 1
Divide by Largest Power of the Variable
T#2
Use the Common Denominator
Rationalize the Numerator (or Denominator)
RATIONALIZE THE DENOMINATOR
Limits at Infinity: Horizontal Asymptotes
Find the horizontal asymptote for the graph of f(x) =
Continuity • Intuitively, a function is said to be continuous if we can draw a graph of the function with one continuous line. I. e. without removing our pencil from the graph paper.