Limits of Rational Functions - Problem 2

I’m finding the limit of another rational function. This time I’m finding the limit as x approaches negative infinity. You shall that the method’s exactly the same as when the limit as x approaches infinity. The first thing I want to do is rewrite the limit.

I want to take this expression and I’m going to multiply the top and bottom by the highest power of x in the denominator. So I have 3x² minus 13x plus 4 over x² minus 9. I multiplied by the highest power of x in the denominator and that’s x². 1 over x², 1 over x².

Now what’s this going to give me? In the numerator I’m going to get 3x² times 1 over x² to 3. And here I’ll get minus 13 x times 1 over x² is minus 13 over x. and here I get plus 4 over x². In the denominator I get x² times 1 over x², 1 minus 9 over x².

Now you’ll notice that what this method does is it makes it so that every one of these terms either approaches a finite or it goes to zero as x goes to either infinity or negative infinity. It doesn’t actually matter. X is going to negative infinity but still 13 over x is going to zero. 4 over x² is going to zero. The denominators are getting bigger and bigger, 9 over x² is going to zero. What we’re left with is 3 minus zero plus zero over 1 minus zero, 3.

That’s it, don’t forget the trick of multiplying by the reciprocal of the highest power of x in the denominator.