If-the-function-f-x-satisfies-lim-x-1-f-x-2-x-2-1-pi-evaluate-lim-x-1-f-x-

Question Number 27701 by NECx last updated on 13/Jan/18

I f t h e f u n c t i o n f (x) s a t i s f i e s

\lim_{x \to 1} \frac{f (x) - 2}{x^{2} - 1} = π, e v a l u a t e \lim_{x \to 1} f (x)

Commented by abdo imad last updated on 21/Jan/18

\Leftrightarrow {l i m}_{x \to 1} \frac{f (x) - 2 - π (x^{2} - 1)}{x^{2} - 1} = 0 s o w e m u s t h a v e

{l i m}_{x \to 1} f (x) - 2 - π (x^{2} - 1) = 0 i n o r d e r t o f i n d t h e f o r m \frac{0}{0}

\Rightarrow {l i m}_{x \to 1} f (x) = 2 l e t v e r i f y t h i s n u m b e r b y h o s p i t a l

t h e o r e m {l i m}_{x \to 1} \frac{f^{^{'}} (x) - 2 π x}{2 x} = 0 \Rightarrow {l i m}_{x \to 1} f^{^{'}} (x) = 2 π s o

w e m u s t h a v e {l i m}_{x \to 1} f (x) = 2 a n d {l i m}_{x \to 1} f^{^{'}} (x) = 2 π .

Answered by mrW2 last updated on 13/Jan/18

\lim_{x \to 1} f (x) = 2

Commented by NECx last updated on 13/Jan/18

h o w ?

Commented by mrW2 last updated on 14/Jan/18

i f \lim_{x \to 1} f (x) \neq 2,

\lim_{x \to 1} \frac{f (x) - 2}{x^{2} - 1} = \frac{f i n i t e}{0} \to \infty \neq π

Commented by abdo imad last updated on 21/Jan/18

t h i s c o n d i t i o n i s u n s u f f i c i e n t \dots .