On-line Math 21

4.4  l'Hôpital's Rule

Example 1

lim x® 0  sin(3x) tan(4x)

Solution

This limit can be found by more elementary means, but it certainly is easy using l'Hôpital's rule:

¹

lim x® 0  sin(3x) tan(4x) = lim x® 0  3cos(3x) 4sec2(4x) = 3 4 ,

since

lim x® 0  cos(3x) = 1 = lim x® 0  sec2(4x).

Footnotes:

¹Many textbooks insist on using a special notation for the equals sign when the step involves l'Hôpital's rule, so that this would look like:

lim x® 0  sin(3x) tan(4x) * = lim x® 0  3cos(3x) 4sec2(4x) ,

or some similar notation. However, no one seems to think it is necessary for a special equals sign to indicate any other particular step has been taken, so it seems odd to do so here. Truth be known, many mathematicians think that l'Hôpital's rule is abused by students, since it is used when more straightforward calculation might be more appropriate. It is this prejudice which impels them to use a special indicator when applying this theorem, as if it were a trick not entirely to be trusted.

Copyright (c) 2000 by David L. Johnson.