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Limits at Infinity 654 Views
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Description:
If you're working on developing a positive self-image, don't ever weigh yourself on planet Deneb-G73. Either the gravitational pull there is a tad wonky, or they really need to get their bathroom scales serviced.
Transcript
- 00:04
Limits at Infinity, a la Shmoop. Astronaut Buzz Goodyear and his crew have
- 00:11
encountered a foreign galaxy…
- 00:13
…and within that galaxy, a unique solar system.
- 00:16
Buzz has calculated equations to represent the gravitational patterns of every planet
- 00:21
in the system.
Full Transcript
- 00:23
Each equation has the format “H” equals f of “t”, where “H” is the height
- 00:28
in meters of the ball above the planet’s surface at any given time “t,”…
- 00:32
…where “t” is the number of seconds passed after the ball first hit the ground.
- 00:38
…that is, he’ll know how the function “H” behaves as “t” approaches infinity…
- 00:48
and beyond The first planet has a gravity equation of
- 00:51
“H” equals 2 “t”.
- 00:53
One second after the ball hits the ground, it’s at 2 feet…
- 00:57
…after five seconds the ball is at 10 feet…
- 00:59
…at a minute, it’s at 120 feet, and so on.
- 01:03
As more and more time passes, that is, as “t” approaches infinity…
- 01:07
…the ball will reach an infinite number of meters above the planet.
- 01:12
Therefore, in math terms, we can say that as “t” approaches infinity, “H” also
- 01:16
approaches infinity.
- 01:18
The general rule with functions like these is that if “t” is raised to a power greater
- 01:22
than zero…
- 01:23
…“H” will approach infinity; The second planet has an equation of “H”
- 01:27
equals 1 over “t.” We can rewrite this equation as t to the negative 1 power.
- 01:34
That means at one second after hitting the ground, the ball would be 1 meter off the
- 01:38
ground.
- 01:39
At five seconds, the ball would be a fifth of a meter off the ground.
- 01:43
As time passes and “t” approaches infinity, “H” gets smaller and smaller and really
- 01:48
really close to 0, but the ball will never actually touch the ground.
- 01:52
So as “t” approaches infinity, H approaches zero.
- 01:56
The general rule with functions like these is that if “t” is raised to a power less
- 02:00
than zero, “H” will approach zero. Now here are three planets in the same orbit,
- 02:06
each with completely different gravitational laws.
- 02:09
We can make general rules for limits that go to infinity based on the powers of the
- 02:13
leading terms of the numerator and denominator.
- 02:16
If the power of the numerator is less than the denominator, then as “t” approaches
- 02:21
infinity., “H” approaches zero .
- 02:23
This is true because the denominator increases faster than the numerator.
- 02:29
Let's test this. The first planet in this orbit has an equation of “H” equals “t”
- 02:33
over… “t” squared plus 1.
- 02:35
If we plug in t equals one, the numerator is one and the denominator is one squared
- 02:40
plus one, which is one plus one, or two.
- 02:45
At t equals ten, H equals ten over ten squared plus one, which is a hundred plus one, or
- 02:51
one hundred one.
- 02:52
If we continue plugging in bigger values for t, the fraction H gets smaller and smaller.
- 02:56
So, as t approaches infinity, H approaches zero, but never actually equals zero.
- 03:02
The second planet in the orbit has an equation of “H” equals 4 “t” squared over … “t”
- 03:08
squared plus 1.
- 03:09
Uh-oh. The powers of the numerator and denominator are the same, so we have to do something different.
- 03:16
Here, we can just take the coefficients and divide them by each other to get the limit.
- 03:21
We can take the leading coefficient of the numerator, four, and divide it by the leading
- 03:25
coefficient of the denominator, one…
- 03:27
…to get that the limit of H as t approaches infinity is four over one, or four.
- 03:34
Finally, the third planet in the orbit has an equation of “H” equals 2 “t” cubed
- 03:39
over… “t” squared plus 1.
- 03:42
Since we are taking the limit as t approaches infinity, we can just divide the leading terms
- 03:45
2 t cubed and t squared to get an accurate approximation of the limit.
- 03:51
We can ignore the plus one because everything besides the leading terms is negligible…
- 03:55
…since the leading terms determine the behavior of a polynomial at infinity.
- 04:00
2 t cubed divided by t squared equals two t.
- 04:03
We can plug in infinity for t to get 2 times infinity.
- 04:07
Infinity isn't really a number, so multiplying it by two just leaves it as infinity.
- 04:13
So, the limit of H as t approaches infinity is also infinity.
- 04:19
As a recap, the general rules for rational functions are:
- 04:24
If the power of the numerator is less than the denominator, then as “t” approaches
- 04:28
infinity., “H” approaches zero
- 04:33
If the power of the numerator and the denominator are the same, then as “t” approaches infinity…
- 04:38
…“H” approaches a constant; specifically, the number achieved when the leading coefficient
- 04:43
of the numerator is divided by that of the denominator.
- 04:46
And finally, if the power of the numerator is greater than the denominator…
- 04:51
…“H” approaches either negative or positive infinity as “t” approaches infinity…
- 04:56
…depending on the signs of the leading coefficients of the numerator and denominator.
- 05:02
There are many rules...but it’s vital to remember that infinity is not a real number,
- 05:05
but a concept, just like forever or eternity.
- 05:08
But if it helps, you can think of infinity as representing a really big, undefined number.
- 05:15
Seems like Buzz is going to have to take his crew elsewhere, though.
- 05:18
Preferably a galaxy where gravity isn’t so bonkers.
- 05:21
This isn’t how he wanted to lose weight.
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