Definite Integration Question 206
Question: $ \underset{x\to a}{\mathop{\lim }}\frac{f(x)-f(a)}{g(x)-g(a)}, $
[EAMCET 1994]
Options:
A) $ \frac{9}{100} $
B) $ \frac{1}{100} $
C) $ \frac{1}{99} $
D) $ \frac{1}{101} $
Show Answer
Answer:
Correct Answer: B
Solution:
$ \underset{n\to \infty }{\mathop{\lim }}\frac{1^{99}+2^{99}+…..+n^{99}}{n^{100}}=\underset{n\to \infty }{\mathop{\lim }}\sum\limits _{r=1}^{n}{( \frac{r^{99}}{n^{100}} )} $
$ =\underset{n\to \infty }{\mathop{\lim }}\frac{1}{n}\sum\limits _{r=1}^{n}{{{( \frac{r}{n} )}^{99}}=\int_0^{1}{x^{99}dx=[ \frac{x^{100}}{100} ]_0^{1}=\frac{1}{100}.}} $
BETA
