Arranging a fraction to simplify it

In summary, you are trying to solve a limit problem involving radicals, but you get lost and make a mistake.
  • #1
LearninDaMath
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Homework Statement

[itex]\stackrel{lim}{x\rightarrow}∞[/itex] [itex]\sqrt{x^2+x}-x[/itex]
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'hopitals to: [itex]\frac{x}{\sqrt{x^2+x}+x}[/itex]
I get: [itex]\frac{1}{\frac{x+1}{x+\sqrt{x}}+1}[/itex]
But don't know how to get to: [itex]\frac{1}{\sqrt{1+\frac{1}{x}}+1}[/itex] ?
And the final solution simplified: 1/2
 
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  • #2
LearninDaMath said:
Applying L'hopitals to: [itex]\frac{x}{\sqrt{x^2+x}+x}[/itex]

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 [itex]\sqrt{x^2}[/itex] is not x to the power of 2.
 
  • #3
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'Hopitals, my professor wants us to use L'Hopitals 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:


[tex]\frac{1}{\frac{x+1}{x+\sqrt{x}}+1}[/tex]


to this:

[tex]\frac{1}{\sqrt{1+\frac{1}{x}}+1}[/tex]


What algebra rules/concepts should I know/apply here?
 
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  • #4
LearninDaMath said:

Homework Statement




[itex]\stackrel{lim}{x\rightarrow}∞[/itex] [itex]\sqrt{x^2+x}-x[/itex]
What's the deal with this one? You don't do anything with this one.
LearninDaMath said:
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'hopitals to: [itex]\frac{x}{\sqrt{x^2+x}+x}[/itex]



I get: [itex]\frac{1}{\frac{x+1}{x+\sqrt{x}}+1}[/itex]
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) x + √x !
This mistake indicate that you don't understand the properties of radicals.
There is NO property that says √(a + b) = √a + √b.

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.
LearninDaMath said:
But don't know how to get to: [itex]\frac{1}{\sqrt{1+\frac{1}{x}}+1}[/itex] ?



And the final solution simplified: 1/2
 
  • #5
in you r question [itex]\lim_{x \to +\infty} \sqrt{x^2+x}-x[/itex]

i don't think there id any need to apply L'hopitals to the equation at all...

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

[tex]\frac{x}{\sqrt{x^2(1+\frac{1}{x})}+x}[/tex]

take the [itex]x^2[/itex] out of the root.

cancel out the top and bottom [itex]x's[/itex]

then apply the fact that [itex]\lim_{x \to +\infty}\frac{1}{x} = 0[/itex]
 
  • #6
Thanks Mark44! Appreciate your feedback. Thanks the-ever-kid, thank you as well.
 

FAQ: Arranging a fraction to simplify it

1. How do you simplify a fraction?

To simplify a fraction, you need to find the greatest common factor (GCF) of the numerator and denominator. Then, divide both the numerator and denominator by the GCF. This will give you the simplified fraction.

2. Can all fractions be simplified?

Yes, all fractions can be simplified as long as the numerator and denominator have a common factor.

3. What is the difference between simplifying and reducing a fraction?

Simplifying a fraction means finding the smallest possible equivalent fraction. Reducing a fraction means dividing both the numerator and denominator by a common factor to make the numbers smaller.

4. Is there a specific order to simplify a fraction?

Yes, there is a specific order to simplify a fraction. First, find the GCF of the numerator and denominator. Then, divide both numbers by the GCF. Finally, simplify the resulting fraction if possible.

5. Can fractions with variables be simplified?

Yes, fractions with variables can be simplified using the same steps as simplifying fractions with numbers. Just remember to combine like terms before finding the GCF.

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