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How do you find the limit of #(ln (ln (x) ) ) / ( ln (x) ) # as x approaches #1#?

Answer 1

Ezekiel Alexander

#+-oo#

Use limit #f(x)/g(x)# = limit#( f'(x))/(g'(x))#.

As #xto1, (ln(ln(x)))/(ln(x))to lim ((ln(ln(x)))')/((ln(x))')#

#= lim ((1/(ln x))(1/x))/(1/x)= lim 1/(ln x)=1/0=+-oo#

Note that the left and the right limits of 1/ln x, as #xto1, are +-oo#. .

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Answer 2

Samantha Adkison

To find the limit of (ln(ln(x))) / (ln(x)) as x approaches 1, we can use L'Hôpital's Rule. Taking the derivative of the numerator and denominator separately, we get:

d/dx (ln(ln(x))) = 1 / (x * ln(x)) d/dx (ln(x)) = 1 / x

Now, we can evaluate the limit by substituting x = 1:

lim(x→1) (ln(ln(x))) / (ln(x)) = lim(x→1) (1 / (x * ln(x))) / (1 / x) = lim(x→1) (1 / (x * ln(x))) * (x / 1) = lim(x→1) 1 / ln(x) = 1 / ln(1) = 1 / 0 = undefined

Therefore, the limit of (ln(ln(x))) / (ln(x)) as x approaches 1 is undefined.

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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.