Question

Ask your teacher find the limit. use l'hospital's rule where appropriate. if there is a more elementary method, consider using it. lim x→1 [ln(x5 − 1) − ln(x3 − 1)]

Answer 1

ln(5/3)

Explanation:

The desired limit represents the logarithm of an indeterminate form, so L'Hopital's rule could be applied. However, the logarithm can be simplified to a form that is not indeterminate.

Limit

We can cancel factors of (x-1), which are what make the expression indeterminate at x=1. Then the limit can be evaluated directly by substituting x=1.

`\diplaystyle \lim\limits_(x\to1){(\ln(x^5-1)-\ln(x^3-1))}=\lim\limits_(x\to1)\ln{\left((x^5-1)/(x^3-1)\right)}\\\\=\lim\limits_(x\to1)\ln\left((x^4+x^3+x^2+x+1)/(x^2+x+1)\right)=\ln{(5)/(3)}`