evaluate : x^(1/lnx)
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Step-by-step explanation:
Explanation:
Let us first plug in
0
right away, keeping in mind that we're approaching from the right side.
lim
x
→
0
+
x
1
ln
x
=
0
1
−
∞
=
0
0
This is an indeterminate form, so we must use l'Hospital's rule.
This function does not appear to be in rational function form,
f
(
x
)
g
(
x
)
; however, a little manipulation will yield that form.
In fact, the following route is usually the one that should be taken when dealing with the indeterminate limit of a function to the power of a function:
Let
y
=
x
1
ln
x
Then,
ln
y
=
ln
(
x
1
ln
x
)
(Applying the logarithm to both sides)
ln
y
=
1
ln
x
(
ln
x
)
=
ln
(
x
)
ln
(
x
)
As
ln
(
x
a
)
=
a
ln
(
x
)
Simplifying yields
ln
y
=
1
lim
x
→
0
+
ln
y
=
lim
x
→
0
+
1
=
1
So, we have the limit for
ln
y
,
but we really want the limit for
y
.
This is no issue, as
lim
x
→
0
+
y
=
lim
x
→
0
+
e
ln
y
=
e
1
=
e
So,
lim
x
→
0
+
x
1
ln
x
=
e
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