SOLUTION: Okay, so I have looked for a calculus tutor online and I have a test on my homework veryy soon. I really need help. Its a La Hospital's Rule... the problem is.... lim

Algebra ->  Algebra  -> Finite-and-infinite-sets -> SOLUTION: Okay, so I have looked for a calculus tutor online and I have a test on my homework veryy soon. I really need help. Its a La Hospital's Rule... the problem is.... lim       Log On

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Question 170695: Okay, so I have looked for a calculus tutor online and I have a test on my homework veryy soon. I really need help. Its a La Hospital's Rule...
the problem is....
lim [ln(x)]^3/x^2
x->infinity


PLEASE help me:)

Answer by Edwin McCravy(6939) About Me  (Show Source):
You can put this solution on YOUR website!

I can't make an arrow for "approaches" easy using 
the software here, so please interpret %27%27%5Bx%2Cinfinity%5D 
as though it were xinfinity.


lim%5Bx%2Cinfinity%5D%28%28ln%28x%29%29%5E3%2Fx%5E2%29 =

We need to get that exponent 3 outside the limit.
First get it outside the parentheses:

lim%5Bx%2Cinfinity%5D%28matrix%284%2C1%2Cln%28x%29%2C%27--------%27%2C%27%27%2Cx%5E%282%2F3%29%29%29%5E3 =

Then get it outside the limit:

%28lim%5Bx%2Cinfinity%5D%28+matrix%284%2C1%2Cln%28x%29%2C%27--------%27%2C%27%27%2Cx%5E%282%2F3%29%29+%29%29%5E3 =

Now with that 3 exponent outside the limit, we can use 
l'Hopital's rule on the inside without it:

%28lim%5Bx%2Cinfinity%5D%28+matrix%284%2C1%2C1%2Fx%2C%27--------%27%2C%27%27%2C%282%2F3%29x%5E%28-1%2F3%29%29+%29%29%5E3 =

Multiply numerator and denominator both by 3x

%28lim%5Bx%2Cinfinity%5D%28+matrix%284%2C1%2C%281%2Fx%29%283x%29%2C%27--------------%27%2C%27%27%2C%282%2F3%29x%5E%28-1%2F3%29%283x%29%29+%29%29%5E3 =

%28lim%5Bx%2Cinfinity%5D%28+matrix%284%2C1%2C3%2C%27--------%27%2C%27%27%2C2x%5E%282%2F3%29%29+%29%29%5E3=0%5E3=0

Edwin