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Limits at Infinity, a la Shmoop. Astronaut Buzz Goodyear and his crew have

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encountered a foreign galaxy…

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…and within that galaxy, a unique solar system.

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Buzz has calculated equations to represent the gravitational patterns of every planet

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in the system.

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Each equation has the format “H” equals f of “t”, where “H” is the height

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in meters of the ball above the planet’s surface at any given time “t,”…

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…where “t” is the number of seconds passed after the ball first hit the ground.

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…that is, he’ll know how the function “H” behaves as “t” approaches infinity…

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and beyond The first planet has a gravity equation of

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“H” equals 2 “t”.

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One second after the ball hits the ground, it’s at 2 feet…

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…after five seconds the ball is at 10 feet…

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…at a minute, it’s at 120 feet, and so on.

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As more and more time passes, that is, as “t” approaches infinity…

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…the ball will reach an infinite number of meters above the planet.

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Therefore, in math terms, we can say that as “t” approaches infinity, “H” also

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approaches infinity.

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The general rule with functions like these is that if “t” is raised to a power greater

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than zero…

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…“H” will approach infinity; The second planet has an equation of “H”

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equals 1 over “t.” We can rewrite this equation as t to the negative 1 power.

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That means at one second after hitting the ground, the ball would be 1 meter off the

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ground.

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At five seconds, the ball would be a fifth of a meter off the ground.

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As time passes and “t” approaches infinity, “H” gets smaller and smaller and really

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really close to 0, but the ball will never actually touch the ground.

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So as “t” approaches infinity, H approaches zero.

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The general rule with functions like these is that if “t” is raised to a power less

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than zero, “H” will approach zero. Now here are three planets in the same orbit,

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each with completely different gravitational laws.

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We can make general rules for limits that go to infinity based on the powers of the

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leading terms of the numerator and denominator.

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If the power of the numerator is less than the denominator, then as “t” approaches

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infinity., “H” approaches zero .

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This is true because the denominator increases faster than the numerator.

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Let's test this. The first planet in this orbit has an equation of “H” equals “t”

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over… “t” squared plus 1.

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If we plug in t equals one, the numerator is one and the denominator is one squared

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plus one, which is one plus one, or two.

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At t equals ten, H equals ten over ten squared plus one, which is a hundred plus one, or

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one hundred one.

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If we continue plugging in bigger values for t, the fraction H gets smaller and smaller.

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So, as t approaches infinity, H approaches zero, but never actually equals zero.

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The second planet in the orbit has an equation of “H” equals 4 “t” squared over … “t”

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squared plus 1.

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Uh-oh. The powers of the numerator and denominator are the same, so we have to do something different.

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Here, we can just take the coefficients and divide them by each other to get the limit.

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We can take the leading coefficient of the numerator, four, and divide it by the leading

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coefficient of the denominator, one…

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…to get that the limit of H as t approaches infinity is four over one, or four.

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Finally, the third planet in the orbit has an equation of “H” equals 2 “t” cubed

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over… “t” squared plus 1.

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Since we are taking the limit as t approaches infinity, we can just divide the leading terms

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2 t cubed and t squared to get an accurate approximation of the limit.

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We can ignore the plus one because everything besides the leading terms is negligible…

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…since the leading terms determine the behavior of a polynomial at infinity.

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2 t cubed divided by t squared equals two t.

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We can plug in infinity for t to get 2 times infinity.

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Infinity isn't really a number, so multiplying it by two just leaves it as infinity.

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So, the limit of H as t approaches infinity is also infinity.

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As a recap, the general rules for rational functions are:

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If the power of the numerator is less than the denominator, then as “t” approaches

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infinity., “H” approaches zero

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If the power of the numerator and the denominator are the same, then as “t” approaches infinity…

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…“H” approaches a constant; specifically, the number achieved when the leading coefficient

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of the numerator is divided by that of the denominator.

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And finally, if the power of the numerator is greater than the denominator…

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…“H” approaches either negative or positive infinity as “t” approaches infinity…

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…depending on the signs of the leading coefficients of the numerator and denominator.

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There are many rules...but it’s vital to remember that infinity is not a real number,

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but a concept, just like forever or eternity.

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But if it helps, you can think of infinity as representing a really big, undefined number.

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Seems like Buzz is going to have to take his crew elsewhere, though.

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Preferably a galaxy where gravity isn’t so bonkers.

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This isn’t how he wanted to lose weight.

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