1. L'Hopital/limits question

Here's the question:
lim(x-->infinity) $\displaystyle x/((1+x^2)^(1/2))$

One approach (that Wolfram Alpha used):
=square root of lim $\displaystyle x^2/(1+x^2) = 1$
(using L'hopital)

Another approach:
Using L'hopital: $\displaystyle =((1+x^2)^(1/2))/x$
which is our original question upside down.
Does this prove the limit is 1.

ie. if limit a/b = limit b/a does this mean the limit must equal 1?

Thanks

2. $\displaystyle \[\dfrac{x}{{\sqrt {1 + x^2 } }} = \dfrac{1}{{\sqrt {\frac{1}{{x^2 }} + 1} }}$

3. No. Try to do the same to f(x)=x/e^x (for your first question!)

4. Originally Posted by durrrrrrrr
ie. if limit a/b = limit b/a does this mean the limit must equal 1?
If both limits exist

limit a/b = 1/(limit b/a)

So limit a/b could also be -1.

5. Originally Posted by Plato
$\displaystyle \[\dfrac{x}{{\sqrt {1 + x^2 } }} = \dfrac{1}{{\sqrt {\frac{1}{{x^2 }} + 1} }}$
It should be:

$\displaystyle f(x)=\dfrac{x}{{\sqrt {1 + x^2 } }} =\begin{Bmatrix} \dfrac{1}{{\sqrt {\frac{1}{{x^2 }} + 1} }} & \mbox{ if }& x> 0\\-\dfrac{1}{{\sqrt {\frac{1}{{x^2 }} + 1} }} & \mbox{if}& x<0\end{matrix}$

So,

$\displaystyle \displaystyle\lim_{x \to{+}\infty}{f(x)}=1,\quad \displaystyle\lim_{x \to{-}\infty}{f(x)}=-1$

and does not exist

$\displaystyle \displaystyle\lim_{x \to{}\infty}{f(x)}$

Fernando Revilla

6. $\displaystyle \displaystyle\lim_{x\to\infty}\frac{x}{\sqrt{x^2+1 }}\Rightarrow\lim_{x\to\infty}\frac{x}{\sqrt{x^2}} \Rightarrow\lim_{x\to\infty}\frac{x}{\pm x}$

$\displaystyle \displaystyle\lim_{x\to\infty}\frac{x}{x}=1 \ \mbox{and} \ \lim_{x\to\infty}\frac{x}{-x}=-1$

7. Because the OP is $\displaystyle \displaystyle \lim _{x \to \infty } \frac{x}{{\sqrt {1 + x^2 } }}$, we can assume that $\displaystyle x>0$

8. Originally Posted by Plato
Because the OP is $\displaystyle \displaystyle \lim _{x \to \infty } \frac{x}{{\sqrt {1 + x^2 } }}$, we can assume that $\displaystyle x>0$
No, you can't assume that. The formula that defines $\displaystyle f$ has sense on $\displaystyle \mathbb{R}$ .When a function is given by a formula an the domain is not specified it is an universal convenious to apply the rule of the maximum domain i.e, $\displaystyle D(f)=\mathbb{R}$, and topologically $\displaystyle \infty\neq +\infty$ .

Fernando Revilla

9. Originally Posted by Plato
Because the OP is $\displaystyle \displaystyle \lim _{x \to \infty } \frac{x}{{\sqrt {1 + x^2 } }}$, we can assume that $\displaystyle x>0$
If we want to assume x>0, then shouldn't the limit be written as $\displaystyle \displaystyle\lim_{x\to\infty^+}\frac{x}{\sqrt{x^2 +1}}\mbox{?}$

10. Originally Posted by dwsmith
If we want to assume x>0, then shouldn't the limit be written as $\displaystyle \displaystyle\lim_{x\to\infty^+}\frac{x}{\sqrt{x^2 +1}}\mbox{?}$

Right.

Fernando Revilla

11. Originally Posted by FernandoRevilla
Hence, the limit DNE since the (+) is infinity and (-) is negative infinity.

12. Originally Posted by dwsmith
Hence, the limit DNE since the (+) is infinity and (-) is negative infinity.
Right.

Fernando Revilla

13. Originally Posted by FernandoRevilla
No, you can't assume that. The function $\displaystyle f$ is defined on $\displaystyle \mathbb{R}$ .When a function is given by a formula an the domain is not specified it is an universal convenious to apply the rule of the maximum domain i.e, $\displaystyle D(f)=\mathbb{R}$, and topologicaly $\displaystyle \infty\neq +\infty$ .
With all due respect, I think you have no idea about how this material is currently being taught. What you have written above is not consistent with any standard calculus text book in use today. I have taught calculus classes since 1964.

14. Does not the notation

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

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

In which case

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

requires x<0 and approaching $\displaystyle -\infty$

15. Originally Posted by Plato
With all due respect, I think you have no idea about how this material is currently being taught. What you have written above is not consistent with any standard calculus text book in use today. I have taught calculus classes since 1964.
Out of curiousity you are saying when infinity is printed in a limit in a Calculus text book, the book is implying the positive values only?

Also, 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.

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