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Trig functions can have limits
applied to them just like any other function.
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To help get us started, we have
3 tricks that help simplify limits used on them.
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Officially, these puppies are
called "Trig Limit Theorems"
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Math buzz words strike again!
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#1 |
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The
first one is pretty obvious: |
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In plain English
it says: |
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The limit of a sine or cosine
function as the variable approaches some value
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is the trig function
applied to that value.
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That is, if a stands for any
number we might have. Then:
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| Lim |
Sin X
= Sin a |
and |
Lim |
Cos X
= Cos a |
| X
®
a |
X
®
a |
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This is maybe
kind of a "duh!, " but to do it, |
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we need an official rule
that says we can. |
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The next two are more
interesting.
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Think of them as ways to cancel
trig stuff out of your problem.
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The trick is to
use algebra to arrange your problem to include one of them |
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and ZAP!
out goes the trig. |
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#2 |
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What this one says is that for
really small angles
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the sine is also a really small number
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and as they
get smaller, they get closer to being the same value.
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Something over
itself equals 1.
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#3
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| Lim |
1
- Cos X |
=
0 |
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| X
®
0 |
X |
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This one is
kind of the same deal. |
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But here, as the cosine gets close to 1 the angle gets close to zero |
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but at a slower rate. |
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So while both
the numerator and denominator get close to zero, |
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the numerator gets
there faster. |
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That makes the
limit of the fraction as x approaches zero equal to zero. |
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Example: |
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| Lim |
Sin
X - Cos XSin X |
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| X
®
0 |
X2
Cos X |
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Factor the
numerator: |
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| Lim |
Sin
X(1 - Cos X) |
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| X
®
0 |
X2
Cos X |
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Separating
terms: |
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| Lim |
( |
Sin
X |
x |
1
- Cos X |
x |
1 |
) |
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| X
®
0 |
X |
X |
Cos X |
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Distributing
the "Lim" |
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| Lim |
Sin
X |
x |
Lim |
1
- Cos X |
x |
Lim |
1 |
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| X
®
0 |
X |
X
®
0 |
X |
X
®
0 |
Cos X |
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Using #2 and #3:
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| 1 |
x |
0 |
x |
Lim |
1 |
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| X
®
0 |
Cos
X |
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We COULD figure
out the last part, |
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but since we are multiplying three things
together and one of them is a zero |
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the rest usually won't matter . |
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As long as the
rest of the terms are defined, the answer will be zero. |
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copyright 2005 Bruce
Kirkpatrick |
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