A counterexample to L’Hopital’s rule
EB January 2003
The following is a counterexample to the naive rule that whenever the 0 0 limit off /gexists, so does the limit off /g
f(x) =x+ cos(x) sin(x),
g(x) = exp(sin(x))f(x).
This is from K. Stromberg,Introduction to Classical Real Analysis, page 188. 0 The proof of L’Hopital’s rule uses the hypothesis thatg(x)6= 0 forx6=a 0 00 although it allows limxag(xIn this example,) = 0.f(x) andg(x) both 0 0 vanish at the zeros of cosxratio. Thef /gextends to a smooth function, but is strictly speaking undened at these zeros.