If-f-x-x-2-4-x-2-x-2-3-0-1-find-Alpha-3-Epsalum-0-1-Find-delta-

Question Number 5232 by sanusihammed last updated on 02/May/16

I f f (x) = \frac{x^{2} - 4}{x - 2}

x \to 2

α = 3 . Σ = 0.1 f i n d δ

A l p h a = 3 E p s a l u m = 0.1 F i n d d e l t a

Commented by FilupSmith last updated on 02/May/16

What is the definition of δ ?

Answered by Yozzii last updated on 02/May/16

A c c o r d i n g t o t h e ϵ - δ d e f i n i t i o n o f

t h e l i m i t, t h e l i m i t \lim_{x \to a} f (x) = α

e x i s t s i f, f o r a n y ϵ > 0 t h e r e i s a δ > 0,

s u c h t h a t

∣ f (x) - α ∣< ϵ w h e n e v e r 0 <∣ x - a ∣< δ .

S o u s i n g f (x) = \frac{x^{2} - 4}{x - 2} a n d α = 4 {n o t 3 s i n c e \lim_{x \to 2} f (x) = \lim_{x \to 2} \frac{(x + 2) (x - 2)}{x - 2} = \lim_{x \to 2} (x + 2) = 2 + 2 = 4 \neq 3 .}

\Rightarrow∣ \frac{x^{2} - 4}{x - 2} - 4 ∣=∣ \frac{x^{2} - 4 - 4 x + 8}{x - 2} ∣

=∣ \frac{x^{2} - 4 x + 4}{x - 2} ∣=∣ \frac{{(x - 2)}^{2}}{x - 2} ∣

=∣ x - 2 ∣

S i n c e ∣ f (x) - 4 ∣< ϵ a n d ∣ f (x) - 4 ∣=∣ x - 2 ∣

\Rightarrow∣ x - 2 ∣< ϵ \Rightarrow l e t ϵ = δ . H e n c e, i f

ϵ = 0.1, δ = 0.1 .

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