Thread: Asymptote question?

1. Asymptote question?

Today in math class, we were learning about asymptotes.

In our math book, it has written the following limits for a vertical asymptote:

$\displaystyle \lim_{x\rightarrow 0^+} \frac{1}{x}=+\infty$

and

$\displaystyle \lim_{x\rightarrow 0^-} \frac{1}{x}=-\infty$

He said that these two limits are wrong, because something can't equal infinity.

Is he right? Everywhere I have looked, these limits have been shown where it equals infinity, yet he says that these statements are false, and our mathbook is wrong for writing this.

2. He (whoever he is) might have meant that infinity is not a real number. In that, he's correct. However, to write a limit as the one given as +infinity or -infinity is valid. I quote from wikipedia:
A related concept to limits as x approaches some finite number is the limit as x approaches positive or negative infinity. This does not literally mean that the difference between x and infinity becomes small, since infinity is not a real number; rather, it means that x either grows without bound positively (positive infinity) or grows without bound negatively (negative infinity).
Limit (mathematics) - Wikipedia, the free encyclopedia

3. They're correct. You can see this from the graph. The problem is when you have something like $\displaystyle 0\times\infty \text{ or } \frac{\infty}{\infty}$, which are indeterminate forms. If these forms arise from the limit of a composite function, you can use l'Hopital's rule.

oops, beat me to it, Chop Suey.

4. okay but you are allowed to write those fraction limits, right?

5. Originally Posted by No Logic Sense
okay but you are allowed to write those fraction limits, right?
Yes.

6. But I'm not sure I quite understand the meaning behind the infinity in this context then.

7. Because if, $\displaystyle f(x)=\frac{1}{x} \text{ then: } \lim_{x\to 0^+}\frac{1}{x} = +\infty$ then this means that as we move x infinitesimally close to 0, without ever actually having x=0 (as x 'tends to' 0), then the function, f(x), tends to infinity, which as Chop suey said, "means that f(x) either grows without bound positively (positive infinity) or grows without bound negatively (negative infinity)." The function itself however can never equal infinity because it's not a real number.