Can't find solution to Calculus 8th (Adams, Essex) limit problem I've been sitting here for hours trying to find a solution to his problem: If you have the function g(y), which is the inverse of f(x)=x^x e^(−1)<=x<oo, Show that lim_(y-> oo}(g(y) ln (ln y))/(ln y) = 1

Tamara Bryan

Tamara Bryan

Answered question

2022-07-17

Can't find solution to Calculus 8th (Adams, Essex) limit problem
I've been sitting here for hours trying to find a solution to his problem:
If you have the function g ( y ), which is the inverse of
f ( x ) = x x e 1 x < ,
Show that
lim y g ( y ) ln ( ln y ) ln y = 1
Hint: Start with the equation y = x x and take the ln of both sides twice.

Answer & Explanation

Kendrick Jacobs

Kendrick Jacobs

Beginner2022-07-18Added 16 answers

If we let that y = x x then g ( y ) = x because that g ( x x ) = x by the definition of the problem. Then if we use the hint given by the problem
ln ( y ) = x ln ( x )
So, for e 1 x < 0 < e x x x < we have that 0 < y < so take x > 1 then
lim y + g ( y ) ln ( ln ( y ) ) ln ( y ) = lim x + x ln ( x ln ( x ) ) x ln ( x ) = lim x + ln ( x ln ( x ) ) ln ( x ) = lim x + ( 1 + 1 ln ( x ) ) = 1
Because ln is a crescent - and not bounded above - function we used that
lim x + 1 ln ( x ) = 0

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