Department of Mathematical Sciences
Instructor: Daiva Pucinskaite
Calculus I
March 22, 2016
Name:
Quiz 13
Evaluate the following limit using l’Hopitals’s Rule
ln x
x→1 4x − x2 − 3
lim
Recall: L’Hopital’s Rule
Let f and g be differentiable on an open interval I containing a with
• lim f (x) = 0
x→a
• lim g(x) = 0
x→a
• g 0 (x) 6= 0 when x 6= a, then
lim
x→a
ln x
x→1 4x − x2 − 3
lim
f (x)
f 0 (x)
= lim 0
x→a g (x)
g(x)
=
(ln x)0
x→1 (4x − x2 − 3)0
=
lim
l’Hopitals’s Rule
=
lim
1
x
4 − 2x
1
lim
x→1 4x − 2x2
x→1
lim 1
=
x→1
2
lim (4x − 2x )
1
4 · 1 − 2 · 12
1
2
x→1
=
=
(since x→1
lim (4x − 2x2 ) 6= 0)