Vertical Asymptote: Is f Defined at x=1?

[dkotschessaa]

Homework Statement

True False

If the line x=1 is a vertical asymptote of y = f(x), then f is not defined at 1.

Homework Equations

none

The Attempt at a Solution

I originally believed this was true, but on finding it was false it sought a counter example:

if for example f(x) = 1/x if x != 0
5 if x = 0

Then the function is defined, but the asymptote still is at x=1, correct?

This is very basic - I just want to make sure I understand it thoroughly. Thanks.

 

[Mentallic]

Homework Statement

True False

If the line x=1 is a vertical asymptote of y = f(x), then f is not defined at 1.

Homework Equations

none

The Attempt at a Solution

I originally believed this was true, but on finding it was false it sought a counter example:

if for example f(x) = 1/x if x != 0
5 if x = 0

Then the function is defined, but the asymptote still is at x=1, correct?

This is very basic - I just want to make sure I understand it thoroughly. Thanks.

The vertical asymptote for that example is at x=0.

So yes the function is defined at x=1 since if we plug x=1 into the equation, we get 1. The asymptote isn't at x=1 though.

 

[dkotschessaa]

Thanks Mentallic!

So you're right. Not a great example. So how about [1/(x-1)] - 1 with f(1) = 5 (or some number)

Point being I guess, that a function can still be defined where there is an asymptote.

 

[dkotschessaa]

p.s. Posting a limit problem over in the calc forum, if you're feeling especially helpful today. This question was actually from my calc book.

 

[Mentallic]

Thanks Mentallic!

So you're right. Not a great example. So how about [1/(x-1)] - 1 with f(1) = 5 (or some number)

If you define the function to be defined at x=1, then that's what it's going to be. But the function f(x)=1/(x-1) alone is not defined at x=1.

Point being I guess, that a function can still be defined where there is an asymptote.

As you've done, yes, but the question was implying there are conditions such as the obvious - you can't define it to be defined at that x value

 

[dkotschessaa]

Well, I was just trying to come up with any example that would serve as a situation where 1) - there is an asymptote at some x and
2) the function is defined at x

I'm sure there are other examples.

Thanks again!

 

[Mentallic]

Well, I was just trying to come up with any example that would serve as a situation where 1) - there is an asymptote at some x and
2) the function is defined at x

I'm sure there are other examples.

Thanks again!

Well yes, under a certain set of conditions. The answer to the problem is no however.