(X/X-1)-(1/lnx) limit is equal to 1

Question

Limit x=1 (×/×-1)-(1/lnX)

Answer 1

You mean: limit x→1 and the first term of the expression is probably intended to be x/(x-1), parenthesized so that the denominator has its own parentheses.  And the rule is called l'Hôpital's Rule.

 

The expression must be written with a common denominator:

[xlnx - (x-1)] / [(x-1)lnx] and the first attempt, using substitution x=1, yields 0/0 and therefore it is justified to use l'Hôpital's Rule.

 

This should be enough to get you started; the next step is to take the ratio of the derivatives of the above bracketed expressions, and so forth.

Answer 2

Limit _{x -> 1}[x/(x - 1) - 1/ln(x)]=1/2

 

By L'Hopital rule:

Limit _{x -> 1}[x/(x - 1) - 1/ln(x)]=

=Limit _{x -> 1}[ln(x)-(x - 1)]/[(x-1)ln(x)]=

=Limit _{x -> 1}[1/x - 1]/[ln(x)-(x-1)/x]=

=Limit _{x -> 1}[1-x]/[xln(x)-(x-1)]=

=Limit _{x -> 1}[- 1/[ln(x)-1-1)]=1/2