Limit logarithm $\lim_{n \to \infty}(4n-6)[\ln(2n+5)-\ln(2n-7)]$.

Question

please any advice for this limit? $$\lim_{n \to \infty}(4n-6)[\ln(2n+5)-\ln(2n-7)]$$ Thanks for any advice.

Answer 1

$\ln(2n+5)-\ln(2n-7)=\ln(\frac{2n+5}{2n-7})=\ln(\frac{2n-7+12}{2n-7})=\ln(1+\frac{12}{2n-7})\sim\frac{12}{2n-7}$

So your limit is equivalent to $\frac{12(4n-6)}{2n-7}\sim24$