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

Answer 1

Evelyn Agard

#1#

#lim_(x to oo) ln (x)/ln (3x)#

#lim_(x to oo) ln (x)/(ln 3 + ln x )#

#lim_(x to oo) 1/(ln 3/ (ln x ) + 1 )#

#(lim_(x to oo) 1)/(lim_(x to oo) ln 3/ (ln x ) + 1 )#

#= 1/1 = 1# as #lim_(x to oo) ln (x) = oo#

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Answer 2

Isaiah Allen

To find the limit of ln(x)/ln(3x) as x approaches infinity, we can use the properties of logarithms and the limit rules.

First, we can rewrite ln(x)/ln(3x) as ln(x)/ln(3) + ln(x)/ln(x).

Next, we can simplify ln(x)/ln(3) as 1/ln(3) since ln(x) approaches infinity as x approaches infinity.

Finally, we can simplify ln(x)/ln(x) as 1 since ln(x) approaches infinity as x approaches infinity.

Therefore, the limit of ln(x)/ln(3x) as x approaches infinity is 1/ln(3).

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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.