A) \[\log (ab)\]
B) \[\log (a/b)\]
C) \[\log (b/a)\]
D) \[-\log (a/b)\]
Correct Answer: C
Solution :
The given limit \[L=\lim \left\{ \frac{1}{na}+\frac{1}{na+1}+\frac{1}{na+2}+...+ \right.\]\[\left. \frac{1}{na+n(b-a)} \right\}\] \[=\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{r=0}^{n(b-a)}{\frac{1}{na+r}}\] \[=\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}\sum\limits_{r=0}^{(b-a)n}{\frac{1}{a+r/n}}\] \[=\int_{0}^{(b-a)}{\frac{dx}{a+x}=[\log (a+x)]_{0}^{b-a}}\] \[\left( \frac{r}{n}=x \right)\] \[=\log \,b-\log a\] \[=\log \left( \frac{b}{a} \right)\].
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