Vertical Asymptote: Is f Defined at x=1?

In summary, the statement "If the line x=1 is a vertical asymptote of y = f(x), then f is not defined at 1" is false. A counterexample is given by the function f(x) = 1/(x-1) - 1, where f(1) is defined as 5. This shows that a function can still be defined at a point where there is an asymptote, but under certain conditions.
  • #1
dkotschessaa
1,060
783

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.
 
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  • #2
dkotschessaa said:

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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
dkotschessaa said:
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.

dkotschessaa said:
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 :wink:
 
  • #6
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!
 
  • #7
dkotschessaa said:
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.
 

1. What is a vertical asymptote?

A vertical asymptote is a line on a graph that a function approaches but never touches. It is represented by a vertical line on the x-axis and indicates a point where the function becomes undefined.

2. How do you determine if a function has a vertical asymptote?

To determine if a function has a vertical asymptote at a specific value of x, you can check if the function approaches positive or negative infinity as x approaches that value. If it does, then there is a vertical asymptote at that point.

3. Is a vertical asymptote the same as a vertical line?

No, a vertical asymptote is not the same as a vertical line. A vertical asymptote is a line that a function approaches but never touches, while a vertical line is a line that passes through a specific point on the graph.

4. Can a function have more than one vertical asymptote?

Yes, a function can have multiple vertical asymptotes. This occurs when a function approaches different points on the x-axis where it becomes undefined.

5. How do vertical asymptotes affect the behavior of a function?

Vertical asymptotes can significantly impact the behavior of a function. They indicate where the function becomes undefined, and can also affect the overall shape of the graph. Functions with vertical asymptotes may have sharp turns or discontinuities near those points.

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