Part II: The Answers
188
Answers
101–200
176.
4
Note that substituting the limiting value, 4, into the function 4
2
−
−
x
x
gives you the indeterminate form 0
0
.
To find the limit, first factor the numerator and simplify:
lim
lim
lim
x
x
x
x
x
x
x
x
x
→
→
→
−
−
=
−
(
) +
(
)
−
(
)
=
+
(
)
4
4
4
4
2
2
2
2
2
Then substitute 4 for x:
lim
x
x
→
+
(
) = + =
4
2
2
4 4
177.
1
Note that substituting in the limiting value, 0, gives you an indeterminate form.
For example, as x approaches 0 from the right, you have the indeterminate form ∞ – ∞,
and as x approaches 0 from the left, you have the indeterminate form –∞ + ∞.
To find the limit, first get common denominators and simplify:
lim
lim
(
)
lim
(
)
(
x
x
x
x x
x
x x x
x
x
→
→
→
−
+






=
−
+






=
+
0
2
0
0
1
1
1
1
1
1
1
x x
x x
x
x x
x
x
x
+
−
+






=
+
=
+
→
→
1
1
1
1
1
1
0
0
)
(
)
lim (
)
lim