Essential Knowledge 1.1CP

Essential Knowledge 1.1CP Students will know how to interpret answers to limits which involve 0 and ∞. They will be able to declare them as 1) a number, 2) ∞ or -∞, or 3) indeterminate.

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Introduction

In limits, two new situations arise that you haven't encountered previously, and we need to learn how to work with those situations. First, answers of limits often include an ∞. Second, limits "allow" you to divide by zero. In other words you don't automatically call fractions with a denominator of zero "undefined".

These two situations can produce three types of answers.

• Answers involving ∞

• Answers involving 0

• Answers involving 0 and ∞

For these three types of answers, our answers could simplify to become

• A real number

  • In AP Calculus, basically anything that exists and isn't ∞ or -∞

• ∞ or -∞

• Indeterminate

  • An answer which may be ∞, -∞, or a real number, but which we can't determine as written. We can use a trick or two to figure out the actual answer

Since situations involving 0 and ∞ are what calculus was truly made for, mastering these limits will give you a strong foundation moving forward.

Key ideas

1. Zero isn't actually zero

If you created a zero during the limit evaluating process, don't think of it as a zero until your final answer. Why? Because remember, limits approach a value without actually reaching it. Whatever you did to make that zero, you did something close but not exact. Until you get to your final answer, you should treat all zeros as very, very small numbers. "Infinitely small."

2. Not all zeros are the same

Sure, both and get smaller as x increases, but will always be twice as small. This holds true even as x→∞, because even though both functions approach zero,

will approach zero twice as fast. Therefore and will both approach zero, but will be smaller, and that can be important if those two zeros ever interact.

3. Not all infinities are the same

Sure, both and get bigger as x gets closer to zero, but will always be twice as big. This holds true even as x→0, because even though both functions approach infinity,

will approach infinity twice as fast. Therefore and will both approach infinity, but

will be bigger, and that can be important if those two infinities ever interact.

4. Division Trends

Regard the table below, where I divided 5 by a variety of numbers.

Behold one of the core ideas in math: in division, dividing by smaller numbers tends to give bigger answers. The smaller the number one divides by, the bigger the answer becomes. The opposite holds true for big numbers

In all situations given below, c represents any positive real number.

I. Answers involving ∞

I.A. Dividing a number by infinity:

Using the division trends above, the larger the number you divide by, the bigger the answer becomes. If we divide by the biggest possible number, we should then get the smallest possible answer. Therefore, the answer will be 0. Sort of. Depends on if your c was negative or positive. For example:

If this limit is part of a larger problem, write a 0, but you should think of it as an extremely small positive number until you reach your final answer. If this is your final step, your answer is simply 0.

If this limit is part of a larger problem, write a 0, but you should think of it as an extremely small negative number until you reach your final answer. If this is your final step, your answer is simply 0.

I.B. Multiplying a number by infinity:

You know that multiplying by a big number gives a big answer. Well infinity is the biggest number, so you'd better get the biggest answer. The sign of the answer depends on the sign of c. For example:

(2)(∞) = ∞, while (-6)(∞) = -∞

I.C. Adding and subtracting numbers with infinity:

and

If you have a big number, adding or subtracting a small number won't effect it much. Adding or subtracting from infinity doesn't effect it at all.

2 + ∞ = ∞, -2 + ∞ = ∞, 2 - ∞ = -∞, -2 - ∞ = -∞,

I.D. Adding and subtracting numbers with infinity:

and

If you have a big number, adding or subtracting a small number won't effect it much. Adding or subtracting from infinity doesn't effect it at all.

I.E. Positive infinity as an exponent:

Sure the absolute value of f(x) increases as x increases, but it's flip-flopping between positive and negative. This means f(x) doesn't approach ∞ or -∞, so this limit does not exist.

(-2)^∞ does not exist

I.F. Negative infinity as an exponent:

Just remember that and you'll be fine. Using what we've learned so far

For positive values of c,

, a very small positive number until the final answer.

For negative values of c, which does not exist.

I.G. Infinity divided by infinity:

Classic calculus! And really, anything could happen. We have a "tug of war" between the two infinities. The top infinity wants to make the answer infinity, the bottom infinity wants to make the answer zero. They don't simply cancel each other out because, remember, not all infinities are the same. The outcome depends on which infinity is increasing faster. We'll learn tricks for this later. For now, since

has multiple possible answers, we'll call it indeterminate.

I.H. Infinity multiplied by infinity:

Remember, you can't make infinity any bigger. (∞)(∞) = ∞. As you would expect, (∞)(-∞) = -∞, (-∞)(-∞) = ∞

I.I. Infinity added or subtracted by infinity:

Remember, you can't make infinity any bigger. ∞ + ∞ = ∞. As you would expect, -∞ - ∞ = -∞

On the other hand, ∞ - ∞ is not so obvious. We have a "tug of war" between the two infinities. The first one wants to make a positive infinity, the second one wants to make a negative infinity. They don't simply cancel each other out because, remember, not all infinities are the same. The outcome depends on which infinity is increasing faster. We'll learn tricks for this later. For now, since ∞ - ∞ has multiple possible answers, we'll call it indeterminate.

I.J. One to the Infinity Power:

Kind of a strange case. Why isn't the answer just 1? If you understand this one, you've probably mastered the idea of limits.

In this case, you'll have to realize that not all ones are the same. Let's look at a few examples.

Example 1 (but is it really "1" haha ok calculus jokes)

The answer to this limit is probably what you've been thinking right from the start: 1. That's because the 1 we're starting with is a "real" one, not a 1 being created by a limit. And you can multiply 1 all day long and it will still be 1.

Example 2

The answer to this limit isn't so simple. Although it looks the same as the previous example, remember not all ones are the same. This one is a tiny bit smaller than 1, because it was subtracted by a zero that was a little bigger than 0. Since the 1 in the base is slightly smaller than 1, we have a tug of war between the 1 and the ∞. The 1 wants to get to 1, and make the answer 1. The infinity wants to make the answer zero, because multiplying numbers smaller than 1 over and over leads to zero. They don't simply cancel each other out because, remember, not all ones or infinities are the same. The outcome depends on which is approaching its goal faster. We'll learn tricks for this later. For now, since this 1^∞ has multiple possible answers, we'll call it indeterminate.

The answer to this limit isn't so simple. Although it looks the same as the previous examples, remember not all ones are the same. This one is a tiny bit bigger than 1, because it was added with a zero that was a little bigger than 0. Since the 1 in the base is slightly bigger than 1, we have a tug of war between the 1 and the ∞. The 1 wants to get to 1, and make the answer 1. The infinity wants to make the answer infinity, because multiplying numbers bigger than 1 over and over leads to infinity. They don't simply cancel each other out because, remember, not all ones or infinities are the same. The outcome depends on which is approaching its goal faster. We'll learn tricks for this later. For now, since this 1^∞ has multiple possible answers, we'll call it indeterminate.

But here's something crazy. The answer is actually e, that number you learned in Algebra II, equal to about 2.718, remember that? Interesting...

II. Answers involving zero

II.A. Dividing a Number by Zero

Using the division trends we saw above, the smaller the number one divides by, the bigger the answer becomes. If we divide by the smallest possible number, we should then get the biggest possible answer. Therefore, the answer will be infinite. Sort of. It could be ∞, -∞, nonexistent. Depends on if your zero is slightly positive, slightly negative, or both.

A few examples might make this clearer. You haven't learned these techniques yet officially, but using what you learned in situation I.A above, you should be fine. Remember, not all zeros are the same!

Example 1

This infinity is positive because the 0 is positve. The zero is positive because 5 and ∞ were both positive when you divided them.

Looking at a graph of f(x) = , you can see f(x) goes to infinity as x→∞

Or you could have just said, wait a second, I know fractions,

, which is just a positively sloped line so of course it will go to infinity.

Example 2

This infinity is negative because the 0 is negative. The zero is negative because -5 and ∞ were opposite signs when you divided them.

Looking at a graph of f(x) = , you can see f(x) goes to negative infinity as x→∞

Or you could have just said, wait a second, I know fractions, , which is just a negatively sloped line so of course it will go to negative infinity.

Example 3

As we saw in example 1, if the zero is positive, the limit will be ∞.

As we saw in example 2, if the zero is negative, the limit will be -∞.

So which is it? Well, since x is approaching a real number, it needs to approach from both the left and the right.

Approaching from the left would give a negative zero, resulting in a limit of -∞.

Approaching from the right would give a positive zero, resulting in a limit of ∞.

The left-handed limit and the right-handed limit are not the same number, so this limit actually does not exist.

A graph of f(x) = 3/x makes this clearer

Clearly one can see that as x approaches zero from the left, the line dips down to negative infinity. From the right, curves up to positive infinity.

II.B.

Classic calculus! And really, anything could happen. We have a "tug of war" between the two zeros. The top zero wants to make the answer zero, the bottom zero wants to make the answer infinity. They don't simply cancel each other out because, remember, not all zeros are the same. The outcome depends on which zero is approaching zero faster. We'll learn tricks for this later. For now, since

has multiple possible answers, we'll call it indeterminate.

II.C.

Anything could happen. We have a "tug of war" between the two zeros. The base zero wants to make the answer zero, the exponent zero wants to make the answer 1. They don't simply cancel each other out because, remember, not all zeros are the same. The outcome depends on which zero is approaching zero faster. We'll learn tricks for this later. For now, since

has multiple possible answers, we'll call it indeterminate.

III. Answers involving 0 and ∞ (the ultimate rivalry!)

III.A.

Classic calculus! And really, anything could happen. We have a "tug of war" between the zero and the infinity. The zero wants to make the answer zero, the infinity wants to make the answer infinity. They don't simply cancel each other out because, remember, not all zeros or infinities are the same. The outcome depends on which is approaching its goal faster. We'll learn tricks for this later. For now, since (0)(∞) has multiple possible answers, we'll call it indeterminate.

III.B.

Anything could happen. We have a "tug of war" between the infinity and the zero. The infinity wants to make the answer infinity, the zero wants to make the answer 1. They don't simply cancel each other out because, remember, not all zeros or infinities are the same. The outcome depends on which is approaching its goal faster. We'll learn tricks for this later. For now, since ∞⁰ has multiple possible answers, we'll call it indeterminate.

III.C.

This one is a good reminder to always use your brain. Don't automatically think, "Zero vs. infinity, must be a tug of war!" Here they are working together. The zero is trying to make the answer 0. The infinity wants to make the answer zero, because multiplying numbers smaller than 1 over and over leads to zero. There is no tug of war, because they want the same thing. The answer is simply 0.