find-gt-0-such-that-f-x-1-lt-0-01-when-0-lt-x-2-lt-where-f-x-x-2-5x-6-x-2-hence-use-definition-to-show-that-lim-xtends-2-f-x-1-

Question Number 25814 by rita1608 last updated on 15/Dec/17

f i n d δ > 0 s u c h t h a t ∣ f (x) + 1 ∣< 0.01

w h e n 0 <∣ x - 2 ∣< δ, w h e r e

f (x) = \frac{x^{2} - 5 x + 6}{x - 2}, h e n c e u s e ε_δ

d e f i n i t i o n t o s h o w t h a t

\frac{l i m}{x t e n d s 2} f (x) = - 1

Answered by ajfour last updated on 15/Dec/17

∣ f (x) + 1 ∣=∣ \frac{(x - 2) (x - 3)}{(x - 2)} + 1 ∣< 0.01

\Rightarrow f o r x \neq 2, ∣ x - 3 + 1 ∣< 0.01

o r ∣ x - 2 ∣< 0.01

s o h e r e δ = ϵ = 0.01

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

Commented by kaivan.ahmadi last updated on 15/Dec/17

δ ⩽ ϵ and Max (δ) = 0.01