Which of the following functions f has a removable...

Which of the following functions $ f $ has a removable discontinuity at $ a $? If the discontinuity is removable, find a function $ g $ that agrees with $ f $ for $ x \neq a $ and is continuous at $ a $. (a) $ f(x) = \dfrac{x^4 -1}{x - 1}$, $ a = 1 $ (b) $ f(x) = \dfrac{x^3 - x^2 - 2x}{x - 2} $, $ a = 2 $ (c) $ f(x) = [ \sin x ] $, $ a = \pi $

Which of the following functions $ f $ has a removable discontinuity at $ a $? If the discontinuity is removable, find a function $ g $ that agrees with $ f $ for $ x \neq a $ and is continuous at $ a $.

(a) $ f(x) = \dfrac{x^4 -1}{x - 1}$, $ a = 1 $
(b) $ f(x) = \dfrac{x^3 - x^2 - 2x}{x - 2} $, $ a = 2 $
(c) $ f(x) = [ \sin x ] $, $ a = \pi $

Transcript

00:01 This is problem number forty nine of the stuart calculus eighth edition section two point five.

00:06 Which of the following functions f has a removable disk? continuity at a.

00:10 If the disk continuity is removed, find a function g that agrees with f for x is not equal to king and his continuous eddie for party.

00:21 Our function have his equal to x to the fourth minus one, divided by the quantity x minus one and is equal to one and ever for uh, f f removed removal of this continent thing first, it needs to have a discontinuity within the function itself.

00:42 We see the function and that a nominator has it is continuity.

00:46 Definitely x equals to one because that will make division zero possible, which is something that it's not allowed it for.

00:56 A sequel to one.

00:58 Ah, however, the only way to check if it is removable is to take the limit as x approaches a or one and determine whether islam it exists for movable discontinuity, the limit must exist.

01:15 Who we take the function next to the fourth minutes one over the quantity expense one.

01:20 Ah, we factor the numerator x squared plus one x squared minus one, and we affect their once more as well.

01:37 We did.

01:37 We did this factory.

01:38 And because it's a difference of squares x squared ah, minus one squared.

01:44 This can be simplified a little further hands ex scripts.

01:54 One will deploy it by x plus one times x minus one divided by experience one, the experience one will cancel our limit.

02:09 We're not ah, evaluating for x equal to one.

02:12 We're approaching x equals one and this limit ends up being one squared plus one.

02:18 That's two.

02:19 Not a lot of buying one plus one wishes to or two times two is four.

02:24 So parliament does exist.

02:26 It's equal to four, all right.

02:28 And this means that the dis continuity at x equals to one is actually removable.

02:34 This community and because it's kind of these removal, we can name a function g equal to and we're in the sequel to is actually dysfunction here.

02:49 That results from simplifying the function holdem simple.

02:54 Fine.

02:55 Um, essentially, what is left over? expert plus one multiplied by experts.

03:01 One is the function g.

03:05 That is continuous city now, and that is our answer for party...

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