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OK, |
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When we
have a zero point in the denominator that can't be factored out |
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we
have a vertical asymptote. |
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There can
be horizontal asymptotes too. |
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(Yeah I
know, big surprise given the name of this page) |
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Horizontal
asymptotes happen |
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when the highest power of the variable in the
numerator |
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is the
same as the highest power of the variable in the denominator. |
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Something
like: |
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When the
highest powers in the numerator and denominator are the same, |
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there will be a
horizontal asymptote at the fraction |
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made by the coefficients of the
highest power terms. |
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In this
case, that will be at 3/5. |
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This is
because as X gets very big, |
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the terms
with lower powers of X, and constants, get less and less important. |
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This
probably makes sense just as it is, |
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but math
types LOVE proofs. |
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So here's
a proof of that |
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"as X gets bigger and bigger" horizontal
asymptote thing. |
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We call
it a limit as X approaches infinity. |
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As usual
with these proofs, you can blow this one off if you want. |
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Just
scroll down till I tell you it's over. |
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But
really, this one's not that bad ... |
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Here
goes. |
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Start
with our original equation and multiply by a messy name for 1. |
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F(x)
= |
3X5
+ 2X2 |
x |
1 |
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| X5
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| 5X5
- 6 |
1 |
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| X5
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Multiply
through and simplify: |
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F(x)
= |
3X5
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+ |
2X2 |
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X5
1 |
X5
3
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5X5
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- |
6 |
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X5
1 |
X5
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Now take
the limit of this as X goes to infinity. |
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Remember,
the limit only cares about terms with X in them... |
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Lim |
F(x)
= |
3 |
+ |
2 |
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| Lim |
X3
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| Xое |
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Xое |
5 |
- |
6 |
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| Lim |
X5
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| Xое |
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Lim |
F(x)
= |
3
+ 0 |
= |
3 |
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Xое |
5
- 0 |
5 |
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| OK! |
Proof's
Over! |
Come
back now! |
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The thing
that makes horizontal asymptotes different from vertical asymptotes |
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is that
the graph line can cross a horizontal asymptote. |
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It can't
cross a vertical asymptote. |
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Diagonal
Asymptotes happen when the biggest exponent in the numerator |
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is 1
larger than the biggest exponent in the denominator. |
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For
example: |
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F(X)
= |
4X5
+ 3X2 |
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| 2X4
+ 2X3 |
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When this
happens, |
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the asymptote will be a diagonal line through the origin
((0,0) point) |
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with a
slope equal to the coefficients of the largest X powers |
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in the
numerator and the denominator. |
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So in the
example above, the slope will be 2 (that is 4/2). |
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This
works about the same way as the horizontal asymptote. |
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As the
value of X gets bigger, the terms with smaller powers get less
important. |
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If we
thought of the equation as just the highest power term in the
numerator |
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and the
highest power term in the denominator, it would be: |
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And this
puppy simplifies to: |
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F(X) = 2X
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And this
is actually known as slope intercept form (F(X) = mX + b) |
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Where m
is the slope and b is the intercept. |
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OK, so m
(the slope) is 2, where's b? |
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It's
zero. |
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Remember,
we said that the diagonal asymptote when through the origin (0,0). |
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That's
the intercept! |
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copyright 2005 Bruce Kirkpatrick |
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