MAT 3749 2.4 Team Homework
Names: ______________________________________________________________
The following problems guide you to prove a version of l’Hospital’s Rule without using
Cauchy’s Mean Value Theorem.
Recall that a function f is strictly increasing on an interval I if f  x   f  y 
whenever x  y .
Lemma 1
Suppose f is continuous on  a, b and differentiable on  a, b  . If f   x   0 x   a, b  ,
then f is strictly increasing on  a, b .
Lemma 2
Suppose f and g are continuous on  a, b , differentiable on  a, b  such that
f  a   g  a  and f   x   g   x  x   a, b  . Then f  x   g  x  x   a, b .
Last Edit: 7/29/2017 3:35 AM
1. l’Hospital’s Rule Version 2
Let f and g be differentiable on a neighborhood of a , at which f  a   g  a   0 , and
g  x   0 .
If lim
xa
f  x
f  x
f  x
exists, then lim
exists and equal to lim
.
xa g  x 
xa g   x 
g x
Use    definitions to prove the l’Hospital Rule Version 2.
(a) (4 points) Show that 1  0 such that f and g are continuous on  a, a  1  ,
and differentiable on  a, a  1  .
(b) lim
xa
f  x
f  x
L  .
exists    0 ,  2  0 such that if a  x  a   2 , then
g x
g  x
where, L  lim
x a
f  x
.
g  x
(Nothing to prove here.)
2
(c) (4 points) Prove that   0 ,   0 such that if a  x  a   , then f and g are
continuous on  a, a    , differentiable on  a, a    , and
 L    g  x   f   x    L    g  x  .
3
(d) (8 points) Prove that   0 ,   0 such that if a  x  a   , then
That is, lim
xa
f  x
g  x
exists and equal to lim
xa
f  x
g  x
L  .
f  x
. (Hint: Use lemma 1 and 2.)
g x
4