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How do you find the Limit of #(x-lnx)# as x approaches infinity?

Answer 1

I found #oo#

I tried some simple substitution to try to understand the behaviour of the function. In particular I noticed the if #x# increases then also #ln(x)# increases BUT more slowly! For example: If #x=1,000# then #ln(1,000)=6.9# If #x=1,000,000# then #ln(1,000,000)=13.8# So I concluded that the limit should be #oo# because #x# "wins" over #ln(x)# going towards #oo#.

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Answer 2

Paisley Johnson

To find the limit of (x-lnx) as x approaches infinity, we can use the concept of L'Hôpital's Rule. By applying this rule, we differentiate the numerator and denominator separately and then take the limit as x approaches infinity.

Differentiating the numerator (x) and denominator (lnx) gives us 1 and 1/x, respectively.

Taking the limit as x approaches infinity, we have:

lim(x->∞) (x-lnx) = lim(x->∞) (1) / (1/x)

Simplifying further, we get:

lim(x->∞) (x-lnx) = lim(x->∞) x * x / 1 = ∞

Therefore, the limit of (x-lnx) as x approaches infinity is infinity.

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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.