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Question Number 8628 by sou1618 last updated on 18/Oct/16

i s i t a l w a y s s a t i s f y i n g ?

A = \lim [n \to \infty] \int f (n, x) d x

B = \int \lim [n \to \infty] f (n, x) d x

A = B ? ?

p l e a s e s h o w c o u n t e r e x a m p l e

c h e c k i n g

(1)

f (n, x) = x^{n}, x [0 \to 1]

A = {l i m}_{n \to \infty} \int_{0}^{1} x^{n} d x = {l i m}_{n \to \infty} \frac{1}{n + 1} x^{n + 1} = 0

B = \int_{0}^{1} {l i m}_{n \to \infty} x^{n} d x = \int_{0}^{1} 0 d x = 0

s o A = B

(2)

f (n, x) = {(1 + \frac{x}{n})}^{n}

A = {l i m}_{n \to \infty} \int {(1 + \frac{x}{n})}^{n} d x

= {l i m}_{n \to \infty} \frac{n}{(n + 1)} {(1 + \frac{x}{n})}^{n + 1}

= e^{x}

B = \int {l i m}_{n \to \infty} {(1 + \frac{x}{n})}^{n} d x

= \int e^{x} d x

= e^{x}

s o A = B

. . .

Answered by prakash jain last updated on 18/Oct/16

Limit and integral exhange is NOT

always allowed .

It is permitted if the function converges

uniformly .

123456 may be able provide more inputs .

Commented by sou1618 last updated on 19/Oct/16

t h a n k s .

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