How do I find a "limit inside a limit"?

Question

I've been struggling with calculus all of a sudden, and for some odd reason while trying to find the limit of x - ln x as x approaches infinity, i know that its an indeterminant form of infinity - infinity, but i can't seem to go on any further because there's no denominator. my teacher keeps on saying about limits inside a limit, but for some reason I can't see it. Please help! thanks

Answer 1

limit x - ln x
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= lim x - lim ln x (use derivative rule)
00 00

= lim 1 - lim 1/x
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= 1 - 0

= 0

Answer 2

I figured it out by just comparing it to another limit. Think of the limit of (x - x/2). That's just the limit of x/2, which you get from subtracting the two numbers. This diverges as x goes to infinity. But ln(x) is going to be much less than x/2 as x gets bigger and bigger. So if ln(x) < x/2 for large x, then x - ln(x) > x - x/2. Thus the limit of (x - ln(x)) must diverge too.

Limits of expressions like (x - x/2) also show why taking the derivative of any "infinity minus infinity" expression will not necessarily give you the same limit. Because the derivative of (x - x/2) is just 1/2. That's obviously not the limit of x/2 as x increases indefinitely!

Answer 3

Multiply and divide by x.
lim(x->∞) x - ln x
= lim(x->∞) (1 - ln x / x) (x)
The limit inside a limit here is lim(x->∞) (ln x / x). This is an ∞/∞ indeterminate form, so our good friend L'Hopital comes into play:
lim(x->∞) ln x / x
= lim(x->∞) (1/x)/1
= 0.
So lim(x->∞) x - ln x
= lim(x->∞) (1 - ln x / x) (x)
= lim(x->∞) (1 - 0) x
= ∞.