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Title: The Properties of Limits and the Squeeze Theorem
Description: Explains the Properties of Limits and the Squeeze Theorem
Description: Explains the Properties of Limits and the Squeeze Theorem
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11-11-2011
Properties of Limits
There are many rules for computing limits
...
There are analogous
results for left and right-hand limits; just replace “ lim ” with “ lim+ ” or “ lim− ”
...
x→c
Remember that a polynomial is a function of the form
a 0 + a 1 x + · · · + a n xn ,
where the a’s are numbers
...
Example
...
x→1
• “The limit of a sum is the sum of the limits
...
x→a
x→a
x→a
This equation — and the ones like it that follow — must be interpreted in the right way
...
Example
...
x→3
• “The limit of a product is the product of the limits
...
x→a
x→a
x→a
Again, the statement is true provided that the two limits on the right side are defined
...
”
n
lim f (x)n =
lim f (x)
x→a
x→a
...
x→a
1
Example
...
x→5
You have to be a little careful with quotients to avoid division by zero
...
f (x)
limx→a f (x)
lim
=
, if g(a) = 0
...
(a) If the numerator approaches a nonzero number, the limit is undefined
...
In any case, the statement is only true if lim f (x) and lim g(x) are defined
...
Using the rules for quotients and for polynomials,
x2 + 3
22 + 3
7
=
=
...
A rational function is a polynomial divided by a polynomial
...
Example
...
Reason: The numerator x2 + 3x − 4 approaches 6 (a nonzero number) while the denominator x − 2
approaches 0
...
lim 2
x→2 x + 4x + 4
16
There’s nothing wrong with having 0 on the top of a fraction
...
x→3 x2 − 10x + 21
Example
...
I need to do more work to determine
0
whether the limit is defined
...
x→3 x2 − 10x + 21
x→3 (x − 3)(x − 7)
x→3 x − 7
−4
2
lim
• “Constants pull out of limits
...
Example
...
x→4
x→4
• “The limit of a root is the root of the limit
...
x→a
3
Example
...
x−2
Since I’m taking an odd root, it doesn’t matter whether the function inside the root is approaching a
positive or a negative number
...
Compute lim
x→4
3
x2 − 5x − 2
=
x−2
3
x2 − 5x − 2
x→−1
x−2
lim
=
3
4
−
...
As x → 4, the quantity x2 − 5x − 1 inside the square root approaches 42 − 5 · 4 − 1 = −5
...
Example
...
√
As x → 3, I have x2 − 9 → 0
...
In fact, the
limit is undefined
...
But what happens if x is
less than 3? Suppose x = 2
...
Then
x2 − 9 = (2
...
41 − 9 = −0
...
59
is undefined
...
Hence, the limit is undefined
...
√
x2 − 9 is
The next result is often called the Sandwich Theorem (or the Squeeze Theorem)
...
The Sandwich Theorem is an intuitively obvious result about limits
...
Suppose you know that:
1
...
x→a
2
...
h(x)
g(x)
f(x)
x=a
Then
lim g(x) = L
...
Example
...
And at x = 0, sin is undefined
...
sin(anything) always lies between −1 and 1:
−1 ≤ sin
1
≤ 1
...
x
Now
lim (−x2 ) = 0 and
lim x2 = 0
...
x
1
1
The picture below shows the graphs of x2 , −x2 , and x2 sin
...
15
0
...
05
-1
-0
...
5
-0
...
1
-0
Title: The Properties of Limits and the Squeeze Theorem
Description: Explains the Properties of Limits and the Squeeze Theorem
Description: Explains the Properties of Limits and the Squeeze Theorem