lim x → a root x + root a / x+a
Answers
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0
Answer:
x
→
∞
(
√
x
2
+
x
−
x
)
=
1
2
Explanation:
The initial form for the limit is indeterminate
∞
−
∞
So, use the conjugate.
(
√
x
2
+
x
−
x
)
=
√
x
2
+
x
−
x
1
⋅
√
x
2
+
x
+
x
√
x
2
+
x
+
x
=
x
2
+
x
−
x
2
√
x
2
+
x
+
x
=
x
√
x
2
+
x
+
x
lim
x
→
∞
x
√
x
2
+
x
+
x
has indeterminate form
∞
∞
, but we can factor and reduce.
We know that
√
x
2
=
|
x
|
, so for positive
x
(which is all we are concerned about for a limit as
x
increases without bound) we have
x
√
x
2
+
x
+
x
=
x
√
x
2
√
1
+
1
x
+
x
(for all
x
≠
0
)
=
x
x
√
1
+
1
x
+
x
(for
x
>
0
)
=
x
x
(
√
1
+
1
x
+
1
)
=
1
√
1
+
1
x
+
1
lim
x
→
∞
1
√
1
+
1
x
+
1
=
1
√
1
+
1
=
1
2
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