Arranging a fraction to simplify it

[LearninDaMath]

Homework Statement

\stackrel{lim}{x\rightarrow}∞ \sqrt{x^2+x}-x
I'm taking the limit of a rational function as x approaches ∞. However, within the problem, there is an algebraic arrangement I'm having trouble with. How would I get from the first fraction to the second fraction?
Applying l'hospital's to: \frac{x}{\sqrt{x^2+x}+x}
I get: \frac{1}{\frac{x+1}{x+\sqrt{x}}+1}
But don't know how to get to: \frac{1}{\sqrt{1+\frac{1}{x}}+1} ?
And the final solution simplified: 1/2

 

[jey1234]

Applying l'hospital's to: \frac{x}{\sqrt{x^2+x}+x}

No need to apply L'Hopital Rule. Have you learned of the method where you factor the highest power of x in the denominator from the fraction?

(If not check Example 2: http://tutorial.math.lamar.edu/Classes/CalcI/LimitsAtInfinityI.aspx)

What is your highest power of x in the denominator? (remember that \sqrt{x^2} is not x to the power of 2.

 

[LearninDaMath]

Hi, yes we learned about the method of factoring or simplifying the fraction w/o applying L'Hopital Rule. The square root of x^2 is lxl. I did notice that I could have applied this method, however, I believe since we are on the chapter about l'hospital's, my professor wants us to use l'hospital's rule so that he knows we know how to do it.

Thus, my confusion is not the calculus portion, but the algebra afterwards. Specifically, I don't know how to get from this:


\frac{1}{\frac{x+1}{x+\sqrt{x}}+1}


to this:

\frac{1}{\sqrt{1+\frac{1}{x}}+1}


What algebra rules/concepts should I know/apply here?

 

[Mark44]

Homework Statement

\stackrel{lim}{x\rightarrow}∞ \sqrt{x^2+x}-x

What's the deal with this one? You don't do anything with this one.

I'm taking the limit of a rational function as x approaches ∞. However, within the problem, there is an algebraic arrangement I'm having trouble with. How would I get from the first fraction to the second fraction?



Applying l'hospital's to: \frac{x}{\sqrt{x^2+x}+x}



I get: \frac{1}{\frac{x+1}{x+\sqrt{x}}+1}

This doesn't follow from what you started with. When you differentiated the square root in the denominator, I think this is what you did:

$$d/dx \sqrt{x^2 + x} = \frac{1}{2\sqrt{x^2 + x}}\cdot (2x + 1) $$
So far, so good, but things start to fall apart after this.
$$= \frac{x + 1}{x + \sqrt{x}}$$

1 - minor mistake -- (1/2)(2x + 1) = x + 1/2, not x + 1
2 - serious mistake -- √(x² + x) ≠[/color] x + √x !
This mistake indicate that you don't understand the properties of radicals.
There is NO property that says √(a + b) = √a + √b.[/color]

I don't know for a fact that this was your thinking, but it sure seems like to me.

In any case, it's much simpler to not use L'Hopital's Rule at all. The expression in the denominator of the original limit is $\sqrt{x^2 + x} + x$

Just factor x out of these two terms, and after some simplification, you can take the limit directly.

But don't know how to get to: \frac{1}{\sqrt{1+\frac{1}{x}}+1} ?



And the final solution simplified: 1/2

 

[the-ever-kid]

in you r question \lim_{x \to +\infty} \sqrt{x^2+x}-x

i don't think there id any need to apply l'hospital's to the equation at all...

Hint: write \sqrt{x^2+x} as \sqrt{x^2(1+\frac{1}{x})}

\frac{x}{\sqrt{x^2(1+\frac{1}{x})}+x}

take the x^2 out of the root.

cancel out the top and bottom x's

then apply the fact that \lim_{x \to +\infty}\frac{1}{x} = 0

 

[LearninDaMath]

Thanks Mark44! Appreciate your feedback. Thanks the-ever-kid, thank you as well.