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 a thousand x squared would turn into a larger value it's going to be one million 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 any n is going to be positive infinity if you square a negative thousand a negative a thousand times a negative thousand is positive one 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 in significant 5 and 2x for example let's say if x is negative a thousand two x is negative two thousand but negative x cubed that's going to be positive one billion which is a lot larger than two thousand 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 3 x cubed minus 5 x to the fourth now 3x cube 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 five x to the fourth and so that's going to be negative five times negative infinity to the fourth power to the fourth power any negative number will become positive and negative five 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 one 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 one divided by 100 is point zero one and one divided by a thousand is point zero zero one notice that as x gets larger and larger the value of the whole fraction gets smaller and smaller in fact it is approaching zero so we could say that the limit as x approaches infinity of one over x is zero 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 zero so let's say if we have the limit as x approaches infinity of five x plus two over seven x minus x squared the degree of the denominator is two the degree of the numerator is one so we have a bottom heavy function anytime it's bottom heavy and if you have in a limit at infinity it's going to equal zero 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 zero because it's bottom heavy same is true for this one that's going to be zero and so zero divided by negative one is zero so the limit approaches zero now here's another example let's say if we have the limit as x approaches infinity of eight x squared minus five x 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 could 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 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 seven over x squared now the limit as x approaches infinity for five over x that's going to become zero and for seven over x squared that's gonna be zero as well because it's bottom heavy and so we're gonna have eight divided by four which will give us the answer of two so that's what you can do if you have uh a rational function with the same degree on top and on the bottom so based on that example go ahead and try this one what is 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 three so the answer is going to be negative seven divided by positive fourteen which is negative one half but now let's show the work so because the highest degree is three i'm going to multiply this hop in the bottom by one over x cubed so 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 gonna turn into a zero the constant will remain the same this is bottom heavy two so that's gonna be another zero so it's gonna be zero plus fourteen and that's bottom heavy so that's going to be zero so we're gonna get negative seven divided by fourteen and fourteen you can write it as seven times two at which point you can cancel a seven so the final answer is negative one over two 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 one over x since the highest degree on the bottom is one on top it's two but they're not the same so i'm going to go with one 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 zero and here we can replace this with infinity so six times infinity is simply infinity and five plus infinity is still a large number infinity if you divide infinity by three you're still going to have a large number so any n 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 four x squared plus nine x minus seven 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 cube 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. so this simplifies to negative x cubed over four x squared which you can reduce that to negative three x over four and so the answer is going to be negative three three-fourths times negative infinity which becomes positive infinity and you could check it if you plug in let's say negative 1000 into this equation you should get a very large positive number you