L Hôpital S Rule Appendix

L’Hôpital’s Rule Appendix

Often we do not need to use L’Hôpital’s Rule to solve fractional limits. In your College (or possibly Intermediate) Algebra class you learned while you were curve sketching that to find the end behavior of a function (horizontal asymptotes) you could compare the degree of the numerator and the denominator. It’s important to note that this only applies for END BHEAVIOR, which, in other words, is limits at infinity. This cannot be applied to limits that go to anywhere but infinity. The table below describes the three cases.

Case Example Limit 1. N > D (numerator lim 2x 2  2 x degree is higher than x 1  denominator degree) 2. N = D lim 2x 2  2 x 2 5x 2 1 5 3. N < D lim 2x  2 x 5x 2 1 0

The above examples need some explanation. If the degree of the numerator is greater than that of the denominator (degree = highest power of x) the limit at  is always equal to  . Basically, both the numerator and denominator are increasing, but the numerator increases “faster” allowing the entire function to increase without bound. If the degree of the numerator is less than that of the denominator, the limit always goes to 0. The denominator increases to infinity much faster than the numerator thereby making the entire function go to 0. The last case is when the degrees of the numerator and the denominator are the same. When this happens, the limit as x approaches  is the ratio of the leading coefficients. The leading coefficients are the numbers in front of the highest power of x. So, in the example above the highest powers of x are 2 in both the numerator and the denominator. The leading coefficient in the numerator is 2 and the leading coefficient in the denominator is 5. The ratio of the two would be 2/5 as stated above. This is a simple way to find the limit. However, this is not L’Hôpital’s rule. If your teacher requires you to use L’Hôpital’s Rule, use it and use the above method as a check.