Question

Calculus Question (Limits and Continuity)


I need to understand why this rational function is continuous ?

Answer 1

In this question we can observe that in the denominator, there is a restriction and that is x cannot equal 5/3. Ok, now we can understand that there is a vertical asymptote at x = 5/3.

Question: Does this mean that this rational function in the form of f(x) = P(x) / Q(x) is continuous ?

Answer 2

According to a theorem in my textbook, it says that all rational functions are continuous functions, but we just saw that there is a vertical asymptote at x =5/3, and that creates discontinuity. Am I right ?

Answer 3

yes, the function is continuous for all x values except where denominator equation equals zero

Answer 4

i'm sorry plz ignore the word equation

Answer 5

ok, so you mean there is a disconinuity when the denominaotr equals zero. Doesn't that make the whole function discontinous? Because now the function is not connected and is not united as a whole. Like there is a cut in between a line.

Answer 6

yes the function `f` would be discontinuous at x=3/5

Answer 7

numerator is continuous x^3+2x^2-1
and denominator is continuous 5-3x
and f(x)= (x^3+2x^2-1)/( 5-3x) is continuous for all x values except x=3/5

Answer 8

Ok, so you mean the point of discontinuity at x = 5/3 will not affect the whole continuity of the function. Got it :)

Answer 9

Thanks for your help

Answer 10

*whole continuity of the function *
do you mean f(x) function right ?
x^3+2x^2-1 divided by 5-3x ?

Answer 11

Yes, exactly. In my opinion, the vertical asymptote will not affect the whole continuity, it will only make the function f(x) discontinuous at a specific point x = 5/3.

Answer 12

okay now make sense
i was confused bec you said ` whole function discontinous`
yes right at that point function is discontinous

Answer 13

i thought you were trying to ask at x=3/5 *whole function* discontinuous or just denominator

Answer 14

I was trying to see if with the point of discontinuity at x = 5/3, would the function as a whole stay continuous which is yes. I understood the concept now. Your explanation was helpful.:)

Answer 15

:=))

Answer 16

"According to a theorem in my textbook, it says that all rational functions are continuous functions"

Look at the rule again there is a part missing from yours...

Answer 17

I asked this question back in OS. lol :D

Answer 18

Yes, yes you did

Answer 19

????

Answer 20

why is everyones name who post a question OpenStudy?

Answer 21

OpenStudy user is created to upload all the questions asked back in OpenStudy, so people can have once again access to it.

Answer 22

However, only the answers of the user who asked the question is posted and not the person who answered the question.

Answer 23

ok