As the hint says, consider the two cases x<1 and x>1 separately. In the x<1 case, you want to make sure that $\displaystyle \ln{x}>-\frac{1}{10}$, so you need x > 0.905. In the x>1 case, you want to make sure that $\displaystyle \ln{x}<\frac{1}{10}$, so you need x < 1.105. In the first case, you would need to set $\displaystyle \delta$ to 1 - 0.905 = 0.095 or less, and in the second case, you would need to set $\displaystyle \delta$ to 0.105 or less. Since you want both to be true, choose $\displaystyle \delta=0.095$.
- Hollywood