Department of Mathematical Sciences
Instructor: Daiva Pucinskaite
Calculus I
July 18, 2016
Quiz 13
Evaluate the following limit using l’Hopitals’s Rule
ln x
lim
x→1 7x − x4 − 6
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
f (x)
f 0 (x)
= lim 0
x→a g(x)
x→a g (x)
lim
Since
lim ln x = 0, and
x→1
ln x
x→1 7x − x4 − 6
lim
lim (7x − x4 − 6) = 7 · 1 − 14 − 6 = 0 we have
x→1
=
(ln x)0
x→1 (7x − x4 − 6)0
=
lim
l’Hopitals’s Rule
lim
1
x
7 − 4x3
1
1
= lim
lim
3
x→1 7x − 4x4
x→1 x(7 − 4x )
x→1
=
lim 1
x→1
(since x→1
lim (7x − 4x4 ) = 7 · 1 − 4 · 14 = 3 6= 0)
lim (7x − 4x4 )
x→1
1
=
7 · 1 − 4 · 14
1
=
.
3
————————————————————————————————————————
=
Evaluate the following limit
ln x
x→1 7x − x4 − 3
lim (7x − x4 − 3) = 7 · 1 − 14 − 3 = 3 6= 0 we have
lim
Since
lim ln x = 0, and
x→1
lim
x→1
x→1
ln x
limx→1 ln x
0
=
= = 0.
4
4
7x − x − 3
limx→1 (7x − x − 3)
3