Homework Help: Limit question

1. Sep 16, 2011

vrmuth

i am also stuck with this one
lim(x=>0) ( (x+9)^(1/2)-(x-9)^(1/2))/x , i wanna evaluate this algebraically,can anybody give me a clue

2. Sep 16, 2011

JHamm

Re: Basic Limit Question

A couple of applications of L'Hopitals rule will do it :)

3. Sep 16, 2011

vrmuth

Re: Basic Limit Question

hi jhamm thanks,could you please show how L'hopitals rule will do it , and cann't it be done algebraically?

4. Sep 16, 2011

Hootenanny

Staff Emeritus
As you have written it, the limit does not exist. Have you copied the question correctly?

5. Sep 16, 2011

SammyS

Staff Emeritus
The domain for the function $\displaystyle f(x)=\frac{\sqrt{x+9}-\sqrt{x-9}}{x}$ is [9, ∞) .

So, as Hootenanny wrote, there appears to be something wrong with the problem as you posted it.

6. Sep 17, 2011

vrmuth

yes the limit does exist, thanks

7. Sep 17, 2011

Harrisonized

Are you sure?

Let f(x) = g(x)/h(x).

Then:

lim f(x)
= lim [g(x)/h(x)]
= [lim g(x)] / [lim h(x)]

8. Sep 17, 2011

vrmuth

did you see the function and that x tends to 0 ? :)

9. Sep 17, 2011

Harrisonized

Let lim g(x) = a, where a∈ℂ (that is, a is a complex variable / it admits complex values). Then lim a/h(x) as h(x)→0 = ∞'​, where ∞'​ denotes the complex infinity (which, as its name suggests, doesn't exist on ℝ, the set of real numbers).

I'm telling you, the limit doesn't exist. Also fyi, L'hopital's rule doesn't work all the time.

10. Sep 17, 2011

symbolipoint

vrmuth,
As described, for your given function, as x approaches 0, makes little sense because the domain cannot include any x value less than 9. You could try some algebraic tricks if you like, but you still have your originally given function. Try using a graphing calculator or a graphing program to display how the function looks, and check what you see "as x approaches 0".

11. Sep 18, 2011

vrmuth

actually i 've written "does exist " instead of "doesn't exist" ,sorry,thanks for everybody

12. Sep 18, 2011

Dickfore

13. Sep 18, 2011

vrmuth

14. Oct 14, 2011

vrmuth

can you please show me some example where L'hopital's rule won't work ?

15. Oct 14, 2011

Harrisonized

lim x→∞ x/√(x2+1)

This limit actually came up in my electrostatics exam a few days ago.

16. Oct 14, 2011

Dickfore

Last edited by a moderator: Apr 26, 2017
17. Oct 14, 2011

Harrisonized

That's great. It's obviously 1. -_-

I'm just providing an example of l'Hopital's rule failing for the limit.

Last edited: Oct 14, 2011
18. Oct 14, 2011

vrmuth

wow! its getting reciprocated each time we apply L'hopital rule , then what's the method to find such limits when l'hopital rule doesn't work ?

19. Oct 14, 2011

Dickfore

Let $L \equiv \lim_{x \rightarrow \infty} \frac{x}{ \sqrt{x^2 + 1} }$. This is indeterminate form of the type $\frac{\infty}{\infty}$. Applying the L'Hospital's Rule with:
$$f(x) = x \Rightarrow f'(x) = 1$$
$$g(x) = \sqrt{x^2 + 1} \Rightarrow g'(x) = \frac{1}{2} (x^2 + 1)^{-\frac{1}{2}} 2 x = \frac{x}{\sqrt{x^2 + 1}}$$

Then:
$$\frac{f'(x)}{g'(x)} = \frac{\sqrt{x^2 + 1}}{x}$$

But, notice that this is the reciprocal of the original fraction! So, we have:
$$L = \frac{1}{L} \Rightarrow L^2 = 1 \Rightarrow L = \pm 1$$
The negative limit is impossible since both of the functions are positive. Thus, we are left with $L = 1[/jtex]. So, L'Hospital's Rule does work in this case. 20. Oct 15, 2011 VietDao29 There's one small error in your work: you assume that the limit does exist to apply L'Hopital's Rule, while the fact that this limit does exist, or not, is still unknown. @vrmuth: To solve these types of problem, we often divide both numerator, and denominator by x to the greatest power (in this problem is x). $$\lim_{x \rightarrow +\infty} \frac{x}{\sqrt{x^2 + 1}}$$ $$=\lim_{x \rightarrow +\infty} \frac{\frac{x}{x}}{\frac{\sqrt{x^2 + 1}}{x}}$$ $$=\lim_{x \rightarrow +\infty} \frac{1}{\sqrt{\frac{x^2 + 1}{x^2}}} = ...$$ It should be easy to go from here. Let's see if you can get 1 as the answer. :) Regards, 21. Oct 15, 2011 Dickfore Well, you can look at the use of l'Hospital's Rule as a heuristic guide. It does give a correct solution if we make the assumption that a finite limit exists. For those that require a rigorous proof, one can then go ahead and construct an epsilon-delta proof by using these identities: $$\left\vert \frac{x}{\sqrt{x^2 + 1}} - 1 \right\vert = \left\vert \frac{x - \sqrt{x^2 + 1}}{\sqrt{x^2 + 1}} \right\vert = \left\vert \frac{x - \sqrt{x^2 + 1}}{\sqrt{x^2 + 1}} \cdot \frac{x + \sqrt{x^2 + 1}}{x + \sqrt{x^2 + 1}} \right\vert = \left\vert \frac{x^2 - x^2 - 1}{\sqrt{x^2 + 1} \left( x + \sqrt{x^2 + 1} \right)} \right\vert$$ $$\frac{1}{\sqrt{x^2 + 1} \left \vert x + \sqrt{x^2 + 1} \right \vert} = \frac{1}{\sqrt{x^2 + 1} \left( x + \sqrt{x^2 + 1} \right)}$$ Next, we notice that [itex]\sqrt{x^2 + 1}$ and $x + \sqrt{x^2 + 1}$ are both monotonically increasing, and the following inequality:
$$\sqrt{x^2 + 1} > |x|, \forall x$$
holds (the graph of the hyperbola lies above its asymptotes). For $x > M > 0$, we have the following series of inequalities:
$$\sqrt{x^2 + 1} \left( x + \sqrt{x^2 + 1} \right) > \sqrt{M^2 + 1} \left( M + \sqrt{M^2 + 1} \right) > M (M + M) > 2 M^2$$

For any given $\epsilon > 0$, we can find such an $M > M_{0} > 0$, so that:
$$2 M^2_0 = \frac{1}{\epsilon}$$
$$M_0 = \frac{1}{\sqrt{2 \epsilon}}$$

$$\left( \forall \epsilon > 0 \right) \left(\exists M_0(\epsilon) > 0 \right) \left( \forall x > M > M_{0} \right) \left\vert \frac{x}{\sqrt{x^2 + 1}} - 1 \right \vert \equiv \frac{1}{\sqrt{x^2 + 1} \left( x + \sqrt{x^2 + 1} \right) } < \epsilon$$

Thus, we had constructed a proof that:
$$\lim_{x \rightarrow \infty}{\frac{x}{\sqrt{x^2 + 1}}} = 1$$

22. Oct 16, 2011

vrmuth

yes, i can get , thanks :)