#### To determine

**To evaluate:** the limit** ** limx→1(xx−1−1lnx)

#### Answer

The value of the limit is limx→1(xx−1−1lnx)=12

#### Explanation

The given limit is in indeterminate form, so we have to use l’Hospital’s Rule, and carry out the simplification.

Rewriting the limit and using l’Hospital’s Rule

limx→1(xx−1−1lnx)=limx→1(xlnx−(x−1)(x−1)lnx) (00Form)=limx→1(x×1x)+lnx−1(x−1)×1x+lnx=limx→1lnxlnx+1−1x=limx→11x1x+1x2=11+1=12

**Conclusion:** limx→1(xx−1−1lnx)=12