3. (21 pts) Evaluate the limit or say that the limit does not exist.
If a limit does not exist, explain why. Show your work.
You may NOT use a table of values, a graph,
or L'Hospitals's Rule to justify your answer.
(Note: Possible answers include + or -..)
(a) (7 pts)
x2 - 2x - 3 -
lim
x+3 Vx+1 - 2
x2 - 2x - 3 Vx+ 1 + 2
lim
x+3 Vx+1 -2 Vx+1 +2
(x - 3) (x + 1) (Vx+
= lim
x+3
x + 1 - 4
= lim
x+3
(x - 3) (x + 1) (Vx+1 +2).
-- x-3
= lin (x + 1) (Vx+1 +2) = (3 + 2) (V3+1 +2) =
= 4 (V4 +2) = 16
(b) (7 pts)
lim xsin (In x)
X
>0+
NOTE:
We cannot apply the Product Law,
since lim sin (1n x) DOES NOT EXIST.
On the other hand,
-1s sin (in x) s 1
and , therefore,
-xs x sin (in x) sx, (since x > 0).
Since lim X = lim (-X) = 0,
x0+
X0
the inequality above implies that
lim xsin (in x) = 0, by the Squeeze Theorem .
x0+
EXPLANATION :
Since lim ln x = -0,
x →0+
sin (lnx) oscillates between 1 and -1 as x + 0*,
and, therefore
lim sin (In x) DOES NOT EXIST.
x
>0+
(c) (7 pts)
in
x+2-
(lnx) +In 20
-=
-00
(x - 2) 70-