Limits at Infinity

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.

AP CalculusLimits
LanguageEnglish Language
MathCalculus

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.