Let’s find the limit of a rational function. Here’s an example. Find the limit as x approaches infinity of 10x² minus 9x minus 9 over x². The key to finding limits as x approaches infinity or as x approaches negative infinity is to rewrite the function in a form where it’s easy to see what the values are approaching.
I use this trick of multiplying the top and bottom by 1 over the highest power of x in the denominator. Let me show you. 1 over x², 1 over x², x² is the highest power of x in the denominator. So I multiply the top and bottom by 1 over x². Watch what happens. You get limit as x approaches infinity and this 1 over x² distributes over the top. You get 10x² times 1 over x², 10. 9x times 1 over x² is 9 over x and 9 over x²? In the denominator you get x² times 1 over x² which is one and so all you have to do is ask yourself what happens as x goes to infinity to 9 over x and to 9 over x².
By the theorem we’ve developed in the previous episode, these go to zero. And so what are you’re left with? 10 minus zero minus zero over 1. This limit is 10 and that’s it. So remember the trick. When you’re finding limits as x approaches infinity or negative infinity, multiply the top and bottom by 1 over the highest power of x in the denominator.