Math Help - L'Hopital/limits question

1. Originally Posted by durrrrrrrr
Here's the question:
lim(x-->infinity) $x/((1+x^2)^(1/2))$

One approach (that Wolfram Alpha used):
=square root of lim $x^2/(1+x^2) = 1$
(using L'hopital)
Well apparently Wolfram Alpha thinks infinity means positive infinity

But it sounds like the limit notation is still not quite standardized, so we should use the most unambiguous one where possible.

2. Originally Posted by Plato
With all due respect, I think you have no idea about how this material is currently being taught.

Well, if you teach $\infty=+\infty$, nothing to say. You are right with that writing convention.

What you have written above is not consistent with any standard calculus text book in use today. I have taught calculus classes since 1964.

That is not my problem. My reasons are scientific not related to habits. Besides a lot of books clearly distinguish between $\infty$ and $+\infty$ .

Fernando Revilla

3. Originally Posted by dwsmith
Out of curiosity you are saying when infinity is printed in a limit in a Calculus text book, the book is implying the positive values only?
I am indeed say exactly that.
The symbol $\displaystyle \lim _{x \to \infty }$ implies that if $N\in \mathbb{Z}^+$ then $x\ge N$.
Surely, that implies that $x>0$.

Originally Posted by dwsmith
my Calc text book isn't written in such a manner. If there isn't a + or -, the it is the overall limit. My book is very good about specifying direction.
Out of curiosity, just what calculus text book did you use?

4. Originally Posted by Plato
I am indeed say exactly that.
The symbol $\displaystyle \lim _{x \to \infty }$ implies that if $N\in \mathbb{Z}^+$ then $x\ge N$.
Surely, that implies that $x>0$.

Out of curiosity, just what calculus text book did you use?
I haven't used it in a while but it is by Ron Larson, Robert Hostetler, and Bruce Edwards.

5. Originally Posted by Archie Meade
Does not the notation

$\displaystyle\lim_{x \to a^{+}}$

denote the limit as "x" approaches "a" from the right?

Right.

Fernando Revilla

6. Originally Posted by dwsmith
I haven't used it in a while but it is by Ron Larson, Robert Hostetler, and Bruce Edwards.
I am not surprised because Bruce Edwards is at the University of Florida.
I know the book well. Have used it many times. But your memory faulty.
In section 5 of chapter 3, there is a detailed discussion of limits ‘at infinity’.
In fact they say that $\displaystyle \lim _{x \to \infty }$ means that x increases without bound.

@FernandoRevilla, I did not know that there were any scientific facts

7. Originally Posted by Plato
@FernandoRevilla, I did not know that there were any scientific facts
If you take $\infty=+\infty$ then, nothig to say, I insist that you are completely right, but you are losing information about different kind of limits.

The reason is that $\mathbb {R}\cup \{\infty\}$ is homeomorphic to $S^1$ (the unit circumference) and $(0,1)$ acts as $\infty$. But $\mathbb{R}\cup\{+\infty,-\infty\}$ is homeomorphic to any close interval $[-a,a]\;(a>0)$ . Then $-a$ acts as $-\infty$ and $a$ acts as $+\infty$ .

Fernando Revilla

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