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Prove the statement using the $ \varepsilon $, $ \delta $ definition of a limit.

$ \displaystyle \lim_{x \to 4}\frac{x^2 - 2x - 8}{x - 4} = 6 $

Given $e>0,$ we need $\delta>0$ such that if $0<|x-4|<\delta$, then $\left|\frac{x^{2}-2 x-8}{x-4}-6\right|<\varepsilon \Leftrightarrow$

$\left|\frac{(x-4)(x+2)}{x-4}-6\right|<\epsilon \Leftrightarrow|x+2-6|<\varepsilon \quad[x \neq 4] \Leftrightarrow|x-4|<\varepsilon,$ So choose $\delta=\epsilon,$ Then

\[

0<|x-4|<\delta \Rightarrow|x-4|<\varepsilon \Rightarrow|x+2-6|<\varepsilon \Rightarrow\left|\frac{(x-4)(x+2)}{x-4}-6\right|<\varepsilon[x \neq 4] \Rightarrow

\]

$\left|\frac{x^{2}-2 x-8}{x-4}-6\right| < t .$ By the definition of a limit, $\lim _{x \rightarrow 4} \frac{x^{2}-2 x-8}{x-4}=6$

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we will prove this statement using absolute delta definition of limit and we will be proven that limit. When eggs goes to four, the expression X square minus two x minus eight over X -4 goes to six At least when x close very close to four the Kocian X squared minus two x minus eight over x minus four is close to six. By definition we got approved the following. For any for every positive absent there exist a positive delta search that if The absolute value of X -4 is less than delta and positive only meaning that X is not equal to four then the absolute value of the function X squared minus two, eggs minus eight Over X -4 and that function -6. Is this an absolute Yeah. So they did is to consider any positive. Excellent. And give some way an expression for delta in terms of accident for which this implication or statement is to. So for that we will look at the following, we start in some way looking at the second part of the statement That is absolute value of X where -2, eggs minus eight Over eggs -4 minus six. And that's equal where you do the algebra here, the common denominator will be X -4. So again in the numerator X square minus two, eggs -8 minus six times eggs minus four. And that then is equal to the absolute value of x square -2, eggs -8 -6 x plus 24 over eggs. When is four we simplify the numerator and we get X square minus eight X. Yeah 20 for -8 is plus 16 over x minus four. And now we know that we can factor out this numerator, X squared minus eight eggs plus 16. In fact is equal to X -4 square because X Men four square will be x square here minus two times four X. That is eight eggs Plus four square. That is 16. So this is a perfect square. And If X is different from four, which in fact we had to consider dad in order to right down this fraction here. Yeah, We can simplify the factor X -4 in terminator with the factor expanded for in the denominator and we get only Absolute Value of X -4. And that's just the expression we have here. And what does it mean? What it means is that we can take delta equals absence simply like that. That is this is an observation. The way of having a clue about the value of delta we gotta take in terms of Excellent. And we have found that in this particular case we can simply take delta. It was equal absent because in that way these implications will be automatic. So let's do the proof forward. That is let's start with let's start with absolute positive given because either any absolutely. Um greater than zero then if we take delta equal. Absolutely then yeah. So the value of X square minus two X -8 over X -4 which is a function minus the limit, which is six. This expression here, the operations we did it already. So I'm not gonna redo that but going to put the result of the equality is exactly equal to X -4 Absolute Value. Okay, all the calculations were done here so you can put it all this over here to have the complete calculation in one place only. And then with this now we are going to use the fact that we start with if if to your list and absolute body of explain its form less than delta, then this is what this less than delta equal, absolute. That is jeez nine and six less than absolute that is We took any positive absalom We stated that we take delta equal absolute that way delta is positive. And if we start with this inequality here, we prove that it's true. Also this inequality here in this case was just very simple, is not that way in all cases, but here it was the situation that is we can t delta is actually equal to absolute people. And yeah, then we have proved that the limit of x square minus two x minus eight over x minus four is six. When x approaches super using absolute that definition

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