How do you find the Limit of # (ln x)^2 / (3x )# as x approaches infinity?

Answer 1

0

on inspection you can see that the logarithmic numerator is going to grow more slowly than the denominator

but as its #oo/oo# indeterminate , the easy way to show this is to L'Hopital it
#lim_(x to oo) (ln x)^2 / (3x )#
By Lhopital #= lim_(x to oo) (2 (ln x)*1/x) / (3 )#
#= 2 lim_(x to oo) ( ln x) / (3x )#
which is still #oo/oo# indeterminate so L'Hopital again
#= 2 lim_(x to oo) ( 1/ x) / (3 )#
#= 2 lim_(x to oo) 1 / (3x ) = 0#
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Answer 2

#lim_{x->oo}(log_e x)/(sqrt(x))=0#

#(log_e x)^2/(3x) = 1/3((log_e x)/(sqrt(x)))^2#
Now #e^x# is monotonic so
#lim_{x->oo}(log_e x)/(sqrt(x))equiv lim_{x->oo}(e^{log_e x})/e^{sqrt x} = lim_{x->oo}x/e^{sqrt x} equiv lim_{y->oo}y^2/e^y = 0#

so

#lim_{x->oo}(log_e x)/(sqrt(x))=0#
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Answer 3

To find the limit of (ln x)^2 / (3x) as x approaches infinity, we can use the limit properties and L'Hôpital's rule.

First, we rewrite the expression as (ln x / x) * (ln x / 3).

As x approaches infinity, ln x also approaches infinity.

Using L'Hôpital's rule, we differentiate the numerator and denominator separately.

The derivative of ln x is 1/x, and the derivative of x is 1.

Taking the limit as x approaches infinity, we get (1/x) / 1, which simplifies to 0.

Therefore, the limit of (ln x)^2 / (3x) as x approaches infinity is 0.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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