Question

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

Answer 1

This limit 'creates' the `infty - infty` indeterminate form so the first step should be finding a common denominator.

Explanation:

`lim_(x to 1) (1/ln(x)-1/(x-1))=lim_(x to 1) (x-1-ln(x))/(ln(x)(x-1))=[0/0]`

And now to get rid of `0/0` you can use the de L'Hôspital's Rule which states that when evaluating `0/0` or `infty/infty` indeterminate forms the limit of the quotient stays the same if derivatives of the numerator and denominator (evaluated seperately, not using the formula for the derivative of a quotient) are substituted for the numerator and denominator, respectively (as long as the 'new' limit exists).

So (in this case using the de L'Hôpital's Rule twice):

`lim_(x to 1) (1-1/x)/((x-1)/x +ln(x))=[0/0]=lim_(x to 1) (1/x^2)/(1/x^2 +1/x)=[1/(1+1)]=1/2`