Understanding Limits at Infinity

In this lesson, we're going to talk about how to find a limit at infinity. So what is the limit as x approaches infinity for the function x squared? So if x became very large, what would x squared become? So let's say if x was 1,000. x squared would turn into a larger value. It's going to be 1,000,000. So as x gets very large, this turns into infinity as well. It becomes even larger, which a large number is still infinity. Now what about the limit as x approaches negative infinity of x squared? Well, if we replace x with negative infinity, when you square a negative number, it's going to be a positive number. So in the end, it's going to be positive infinity. If you square a negative 1,000, a negative 1,000 times a negative 1,000 is positive 1 million, so you still get a very large positive number. Now what about the limit as x approaches infinity for positive x cubed? Infinity to the third power is still going to be a large positive number, so that's going to be positive infinity. And as x approaches negative infinity, because it's raised to the third power, it's going to be negative. So your final answer is negative. Infinity. Now let's look at some other examples. What is the limit as x approaches negative infinity of this expression? 5 plus 2x minus x cubed. Now, if you're given a polynomial function, you could... Ignore the insignificant terms. 5 and 2x are insignificant compared to negative x cubed. So this expression is equivalent to the limit as x approaches negative infinity of negative x cubed. When x becomes very large, the other terms are insignificant, 5 and 2x. For example, let's say if x is negative 1,000. 2x is negative 2,000. But negative x cubed, that's going to be positive 1 billion, which is a lot larger than 2,000. So we have negative, and then negative infinity to the third power. Negative infinity to the third power is just negative infinity. And then negative times negative infinity will give you a final answer of positive infinity. And so that's going to be the answer for that problem. Here's another one you could try. What is the limit as x approaches negative infinity of 3x cubed minus 5x to the fourth? Now, 3x cubed is insignificant compared to 5x to the fourth. This has a higher degree. So this expression is equivalent to the limit as x approaches negative infinity of negative 5x to the fourth. to be negative 5 times negative infinity to the fourth power. To the fourth power, any negative number will become positive. And negative 5 times positive infinity is going to be a very large negative number, so we're going to say that's negative infinity. And so that's the answer. Now you need to know how to find the limit at infinity given a rational function. So what is the limit as x approaches infinity for 1 over x? Now you need to know what happens to the value of a fraction whenever the denominator increases in value. So for example, let's say if x is 0.1. Actually no, a large number, let's say like 10. 1 divided by 10 is 0.1. Now, if we increase the value of the denominator to 100, notice that the value of the fraction will become even smaller. 1 divided by 100 is 0.01, and 1 divided by 1,000 is 0.001. Notice that as x gets larger and larger, the value of the whole fraction gets smaller and smaller. In fact, it is approaching 0. So we could say that the limit as x approaches infinity of 1 over x is 0. In fact, the limit... as X approaches any large number for any rational function or a function where it's bottom heavy, where the degree of the denominator exceeds that of the numerator, will always be 0. So let's say if we have the limit as X approaches infinity. of 5x plus 2 over 7x minus x squared. The degree of the denominator is 2. The degree of the numerator is 1. So we have a bottom-heavy function. Anytime it's bottom-heavy, and if you have a limit at infinity, it's going to equal 0. Now, if you want to show your work, here's what you can do. Multiply the top and the bottom by 1 over x squared. So what we're going to have is the limit as x approaches infinity, 5 divided by x plus 2 over x squared. And then this is going to be 7 over x minus 1. So this expression turns into a 0 because it's bottom heavy. Same is true for this one. That's going to be 0. And so 0 divided by negative 1 is 0. So the limit approaches 0. Now here's another example. Let's say if we have the limit as x approaches infinity of 8x squared minus 5x over 4x squared plus 7. So what's the answer? Notice that in this example, the degree of the numerator is the same as that of the denominator. When you see that, you can simply divide the coefficients. So it's going to be 8 divided by 4, which is equal to 2. And so that's the answer for this limit. But if you need to show your work, what we're going to do is multiply the top and the bottom by 1 over x squared. And so we're going to have the limit as x approaches infinity of 8 minus 5 over x divided by 4 plus 7 over x squared. Now, the limit as x approaches infinity for 5 over x, that's going to become 0. And for 7 over x squared, that's going to be 0 as well because it's bottom heavy. And so we're going to have 8. 8 divided by 4, which will give us the answer of 2. So that's what you can do if you have a rational function with the same degree on top and on the bottom. So based on that example, go ahead and try this one. the limit as X approaches negative infinity of 5x minus 7x cubed over 2x squared plus 14x cubed minus 9. So looking at this, the degree of the top and the bottom is 3. So the answer is going to be negative 7 divided by positive 14, which is negative 1 half. But now let's show the work. So because the highest degree is 3, I'm going to multiply the top and the bottom by 1 over x cubed. So what we're going to have is the limit as x approaches negative infinity. This is going to become 5 over x squared minus 7 divided by 2 over x. plus 14 minus 9 over x cubed. So here we have a rational function that's bottom heavy. That's going to turn into a 0. The constant will remain the same. This is bottom heavy, too, so that's going to be... be another 0 so it's gonna be 0 plus 14 and that's bottom heavy so that's gonna be 0 so we're gonna get negative 7 divided by 14 and 14 you can write it as 7 times 2 At which point you can cancel a 7. So the final answer is negative 1 over 2. And so that's how you can calculate the limit of that expression. Now what about a rational function that's top heavy? Let's say this is 5x plus 6x squared divided by 3x minus 8. So what do you do in this case? What I like to do, if you want to do it mentally, a simple way is to remove the insignificant terms. In the numerator, the insignificant term is 5x. 6x squared has much more weight than 5x. On the bottom, 3x is more significant than negative 8. So this expression is equivalent to the limit as x approaches infinity. of positive 6x squared divided by 3x. And then that reduces to 2x. So this is going to be 2 times infinity. So your final answer should be positive infinity. But now let's do this problem another way. So just like we've been doing before, let's multiply the top and the bottom by 1 over x. Since the highest degree on the bottom is 1. On top it's 2, but they're not the same, so I'm going to go with 1 this time. So on top it's going to be 5 plus 6x. And on the bottom, 3 minus 8 over x. So this is a bottom heavy function. When x becomes large, it's going to turn to 0. And here we can replace this with infinity. So 6 times infinity is simply infinity. And 5 plus infinity is still a large number of infinity. If you divide infinity by 3, you're still going to have a large number. So in the end, the final answer is just infinity. So let's do one more example, just in case you found the last example a bit confusing. So this is going to be 5 plus 2x minus 3x cubed over 4x squared. plus 9x minus 7. So once again, we have a function that's top-heavy. Now, what I like to do personally, my way of quickly getting the answer, is to eliminate every term that's insignificant. In the numerator, the most significant term is the 3x cubed. So we can ignore the 5 and the 2x. In the denominator, keep the most significant term, which is 4x squared. So we're going to be left, we can get rid of the 9x and the 7. Thank you. so this simplifies to negative 3x cubed over 4x squared which you can reduce that to negative 3x over 4 So the answer is going to be negative 3 fourths times negative infinity, which becomes positive infinity. And you can check it. If you plug in, let's say, negative 1000 into this equation, you should get a very large positive number.

So let's say if x was 1,000. x squared would turn into a larger value. It's going to be 1,000,000. So as x gets very large, this turns into infinity as well. It becomes even larger, which a large number is still infinity.

Now what about the limit as x approaches negative infinity of x squared? Well, if we replace x with negative infinity, when you square a negative number, it's going to be a positive number. So in the end, it's going to be positive infinity. If you square a negative 1,000, a negative 1,000 times a negative 1,000 is positive 1 million, so you still get a very large positive number. Now what about the limit as x approaches infinity for positive x cubed?

Infinity to the third power is still going to be a large positive number, so that's going to be positive infinity. And as x approaches negative infinity, because it's raised to the third power, it's going to be negative. So your final answer is negative. Infinity.

Now let's look at some other examples. What is the limit as x approaches negative infinity of this expression? 5 plus 2x minus x cubed.

Now, if you're given a polynomial function, you could... Ignore the insignificant terms. 5 and 2x are insignificant compared to negative x cubed. So this expression is equivalent to the limit as x approaches negative infinity of negative x cubed. When x becomes very large, the other terms are insignificant, 5 and 2x.

For example, let's say if x is negative 1,000. 2x is negative 2,000. But negative x cubed, that's going to be positive 1 billion, which is a lot larger than 2,000.

So we have negative, and then negative infinity to the third power. Negative infinity to the third power is just negative infinity. And then negative times negative infinity will give you a final answer of positive infinity.

And so that's going to be the answer for that problem. Here's another one you could try. What is the limit as x approaches negative infinity of 3x cubed minus 5x to the fourth? Now, 3x cubed is insignificant compared to 5x to the fourth.

This has a higher degree. So this expression is equivalent to the limit as x approaches negative infinity of negative 5x to the fourth. to be negative 5 times negative infinity to the fourth power. To the fourth power, any negative number will become positive.

And negative 5 times positive infinity is going to be a very large negative number, so we're going to say that's negative infinity. And so that's the answer. Now you need to know how to find the limit at infinity given a rational function. So what is the limit as x approaches infinity for 1 over x? Now you need to know what happens to the value of a fraction whenever the denominator increases in value.

So for example, let's say if x is 0.1. Actually no, a large number, let's say like 10. 1 divided by 10 is 0.1. Now, if we increase the value of the denominator to 100, notice that the value of the fraction will become even smaller. 1 divided by 100 is 0.01, and 1 divided by 1,000 is 0.001.

Notice that as x gets larger and larger, the value of the whole fraction gets smaller and smaller. In fact, it is approaching 0. So we could say that the limit as x approaches infinity of 1 over x is 0. In fact, the limit... as X approaches any large number for any rational function or a function where it's bottom heavy, where the degree of the denominator exceeds that of the numerator, will always be 0. So let's say if we have the limit as X approaches infinity. of 5x plus 2 over 7x minus x squared.

The degree of the denominator is 2. The degree of the numerator is 1. So we have a bottom-heavy function. Anytime it's bottom-heavy, and if you have a limit at infinity, it's going to equal 0. Now, if you want to show your work, here's what you can do. Multiply the top and the bottom by 1 over x squared.

So what we're going to have is the limit as x approaches infinity, 5 divided by x plus 2 over x squared. And then this is going to be 7 over x minus 1. So this expression turns into a 0 because it's bottom heavy. Same is true for this one. That's going to be 0. And so 0 divided by negative 1 is 0. So the limit approaches 0. Now here's another example.

Let's say if we have the limit as x approaches infinity of 8x squared minus 5x over 4x squared plus 7. So what's the answer? Notice that in this example, the degree of the numerator is the same as that of the denominator. When you see that, you can simply divide the coefficients.

So it's going to be 8 divided by 4, which is equal to 2. And so that's the answer for this limit. But if you need to show your work, what we're going to do is multiply the top and the bottom by 1 over x squared. And so we're going to have the limit as x approaches infinity of 8 minus 5 over x divided by 4 plus 7 over x squared. Now, the limit as x approaches infinity for 5 over x, that's going to become 0. And for 7 over x squared, that's going to be 0 as well because it's bottom heavy.

And so we're going to have 8. 8 divided by 4, which will give us the answer of 2. So that's what you can do if you have a rational function with the same degree on top and on the bottom. So based on that example, go ahead and try this one. the limit as X approaches negative infinity of 5x minus 7x cubed over 2x squared plus 14x cubed minus 9. So looking at this, the degree of the top and the bottom is 3. So the answer is going to be negative 7 divided by positive 14, which is negative 1 half. But now let's show the work.

So because the highest degree is 3, I'm going to multiply the top and the bottom by 1 over x cubed. So what we're going to have is the limit as x approaches negative infinity. This is going to become 5 over x squared minus 7 divided by 2 over x.

plus 14 minus 9 over x cubed. So here we have a rational function that's bottom heavy. That's going to turn into a 0. The constant will remain the same. This is bottom heavy, too, so that's going to be...

be another 0 so it's gonna be 0 plus 14 and that's bottom heavy so that's gonna be 0 so we're gonna get negative 7 divided by 14 and 14 you can write it as 7 times 2 At which point you can cancel a 7. So the final answer is negative 1 over 2. And so that's how you can calculate the limit of that expression. Now what about a rational function that's top heavy? Let's say this is 5x plus 6x squared divided by 3x minus 8. So what do you do in this case?

What I like to do, if you want to do it mentally, a simple way is to remove the insignificant terms. In the numerator, the insignificant term is 5x. 6x squared has much more weight than 5x. On the bottom, 3x is more significant than negative 8. So this expression is equivalent to the limit as x approaches infinity.

of positive 6x squared divided by 3x. And then that reduces to 2x. So this is going to be 2 times infinity.

So your final answer should be positive infinity. But now let's do this problem another way. So just like we've been doing before, let's multiply the top and the bottom by 1 over x.

Since the highest degree on the bottom is 1. On top it's 2, but they're not the same, so I'm going to go with 1 this time. So on top it's going to be 5 plus 6x. And on the bottom, 3 minus 8 over x.

So this is a bottom heavy function. When x becomes large, it's going to turn to 0. And here we can replace this with infinity. So 6 times infinity is simply infinity.

And 5 plus infinity is still a large number of infinity. If you divide infinity by 3, you're still going to have a large number. So in the end, the final answer is just infinity.

So let's do one more example, just in case you found the last example a bit confusing. So this is going to be 5 plus 2x minus 3x cubed over 4x squared. plus 9x minus 7. So once again, we have a function that's top-heavy. Now, what I like to do personally, my way of quickly getting the answer, is to eliminate every term that's insignificant.

In the numerator, the most significant term is the 3x cubed. So we can ignore the 5 and the 2x. In the denominator, keep the most significant term, which is 4x squared.

So we're going to be left, we can get rid of the 9x and the 7. Thank you. so this simplifies to negative 3x cubed over 4x squared which you can reduce that to negative 3x over 4 So the answer is going to be negative 3 fourths times negative infinity, which becomes positive infinity. And you can check it. If you plug in, let's say, negative 1000 into this equation, you should get a very large positive number.

Transcript for:Understanding Limits at Infinity

Transcript for:
Understanding Limits at Infinity