find-lim-x-gt-1-x-x-2-dt-ln-t-

Question Number 27792 by abdo imad last updated on 14/Jan/18

f i n d {l i m}_{x - > 1} \int_{x}^{x^{2}} \frac{d t}{l n (t)} .

Commented by abdo imad last updated on 15/Jan/18

i f f c o n t i n u e i n [a, b] a n d g i n t e g r a b l e i n [a, b] \exists c \in] a, b [/

\int_{a}^{b} f (x) g (x) d x = f (c) \int_{a}^{b} g (x) d x s o \exists c_{x} \in] x, x^{2} [/

\int_{x}^{x^{2}} \frac{d t}{l n t} = \frac{1}{l n c} \int_{x}^{x^{2}} d x = \frac{x^{2} - x}{{l n c}_{x}} = \frac{x (x - 1)}{l n (c_{x})} b u t x - > 1 \Leftrightarrow c_{x} - > 1

s o {l i m}_{x - > 1} \int_{x}^{x^{2}} \frac{d t}{l n t} = {l i m}_{c_{x - > 1}} \frac{c_{x} (c_{x}^{} - 1)}{{l n c}_{x}}

{l i m}_{c_{x - > 1}} \frac{1}{\frac{{l n c}_{x}}{c_{x} - 1}} = 1 . (w e c a n c o n f u s e x a n d c_{x} b r c a u s e

x \in V (1) a n d c_{x} \in V (1) .

Commented by prakash jain last updated on 16/Jan/18

understood rest of the solution but

What is V (1) ? is it vicinity of 1 ?

PS : there are symbols in the app like

\to, \infty which you can use press

green \int button on bottom left corner .

Commented by abdo imad last updated on 16/Jan/18

x \in V (1) m e a n t h a t x i s s o n e a r f r o m

1 .