AP Calc 1.5 Notes Limits.notebook
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August 31, 2017
Aug 308:47 AM Sep 74:11 PM
To find the tangent to a curve or the velocity of an object, we now turn our attention to limits in general and numerical and graphical methods for computing them.
Let’s investigate the behavior of the function f defined by f (x) = x2 – x + 2 for values of x near 2.
The following table gives values of f (x) for values of x close to 2 but not equal to 2.
Picking up from 1.4...
Sep 74:17 PM
In fact, it appears that we can make the values of f(x) as close as we like to 4 by taking x sufficiently close to 2We express this by saying:
The limit of f(x) as x approaches 2 is 4
The notation for this is:
Sep 74:20 PM
This says that the values of f (x) tend to get closer and closer to the number L as x gets closer and closer to the number a (from either side of a) but x ≠ a.
An alternative notation:
As x a, f(x) L
Sep 74:22 PM
What is the significance of "but x ≠ a" in the definition of the limit?
What is limx→a f (x) in each of these cases?
limx→a f (x) =L limx→a f (x) =L limx→a f (x) =L
Sep 74:26 PM
2 1 0 1 2 3
1
1
2
3
4
5
x
yGuess the value of
=0.5
AP Calc 1.5 Notes Limits.notebook
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August 31, 2017
Sep 74:47 PM
Now let’s change f slightly by giving it the value 2 when x = 1 and calling the resulting function g:
=0.5 2 1 0 1 2 3
1
1
2
3
4
5
x
y
Sep 74:52 PM
Estimate the value of .
0.166670.16668
0.167
Sep 75:04 PM
The Heaviside step function H is defined by
H (t) approaches 0 as t approaches 0 from the left and H (t) approaches 1 as t approaches 0 from the right.
We indicate this situation symbolically by writing:
The symbol “t → 0–” indicates t approaching 0 from the left.
Likewise, “t → 0+” indicates t approaching 0 from the right.
Sep 75:08 PM
Similarly, if we require that x be greater than a, we get “the righthand limit of f (x) as x approaches a is equal to L” and we write
What's the difference between definitions 2 and 1?
Sep 75:09 PM
By comparing Definition 1 with the definitions of onesided limits, we see that the following is true.
In words:
The limit of f(x) as x approaches a exists if and only if (iff) the left and right side limits are the same
Sep 75:12 PM
The graph of a function g is shown in Figure 10. Use it to state the values (if they exist) of the following:
3
2 2 2
1 DNE
Calculus Democracy?1 2
AP Calc 1.5 Notes Limits.notebook
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August 31, 2017
Sep 75:16 PM Sep 75:18 PM
Again, the symbol ∞ is not a number, but the expression limx→a f (x) = ∞ is often read as
“the limit of f (x), as approaches a, is infinity”
or “f (x) becomes infinite as approaches a”
or “f (x) increases without bound as approaches a”vertical asymptote
Sep 75:21 PM
Similar definitions can be given for the onesided infinite limits:
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Sep 75:24 PM
Homework:
Finish problem set 1.5 in your Booklet