Infinity Limit

[johnjones]

lim x-> infinity sroot(x^2+10x) - x

It's infinity pllus infinity type. I'm not sure what I should do. Thx.

 

[tkhunny]

lim x-> infinity sroot(x^2+10x) - x

It's infinity pllus infinity type. I'm not sure what I should do. Thx.

Try "rationalizing" the numerator. Multiply numerator and denominator by sroot(x^2+10x) + x and see what pops out.

 

[johnjones]

lim x-> infinity sroot(x^2+10x) - x

It's infinity pllus infinity type. I'm not sure what I should do. Thx.

Try "rationalizing" the numerator. Multiply numerator and denominator by sroot(x^2+10x) + x and see what pops out.

lim x -> infinity x^2+10x+x/sroot( x^2+10x) + x

That's what I got so far. Should I divide everything by x now? Should I rationalize again? Difference of squares?

 

[tkhunny]

None of the above. You should do the numerator over again. It went a little off, there.

Also, please provide additional grouping symbols to clarify meaning. I doubt that what you have written is what you intended. Remember the "Order of Operations". It will help you know where to put parentheses.

 

[johnjones]

None of the above. You should do the numerator over again. It went a little off, there.

Also, please provide additional grouping symbols to clarify meaning. I doubt that what you have written is what you intended. Remember the "Order of Operations". It will help you know where to put parentheses.

I don't know how to multiply it out . I wrote sroot(x^2+10x) as (x^2+10x)^2. Then I multipled [{sroot(x^2+10x)} - x] by [{sroot(x^2+10x)} + x] . I expanded, ended up with:


~ ~ ~
[(x^2+10x)^2 - x] * [(x^2+10x) + x]
(x^4+20x^3+100x^2-x)(x^4+20x^3+100x^2+x)


x^8+40x^6+10000x^4+x^5+20x^7+400x^6+2000x^5+20x^4+100x^6+2000x^5+10000x^4+100x^3-x^5-20x^4+200x^4+100x^3.

~ ~ ~

I think there's a more simple way to doing this . Please help. thx :?:

 

[pka]

[√(x<SUP>2</SUP>+10x)−x][(√(x<SUP>2</SUP>+10x)+x)/(√(x^2+10x)+x)]= (x<SUP>2</SUP>+10x)−x<SUP>2</SUP>)/(√(x<SUP>2</SUP>+10x)+x)=(10x)/(√(x<SUP>2</SUP>+10x)+x).
Divide everything by x. Can you see the limit is 5?

 

[tkhunny]

[(x^2+10x)^2 - x] * [(x^2+10x) + x]

First, your square root seems to have disappeared.

Second, the factor was selected because it was easy and useful. You should recognize this structure as a difference of squares. Don't even consider expanding it any more than it is already. What does a difference of squares do? The middle term goes away and you are left with the two squares. Don't algebra yourself to death when it is an eyeball problem.

[(sqrt(x^2+10x))^2 - x] * [(sqrt(x^2+10x)) + x] =
(sqrt(x^2+10x))^2 - x^2 =
x^2 + 10*x - x^2 =
10*x

 

[johnjones]

[√(x<SUP>2</SUP>+10x)−x][(√(x<SUP>2</SUP>+10x)+x)/(√(x^2+10x)+x)]= (x<SUP>2</SUP>+10x)−x<SUP>2</SUP>)/(√(x<SUP>2</SUP>+10x)+x)=(10x)/(√(x<SUP>2</SUP>+10x)+x).
Divide everything by x. Can you see the limit is 5?

Sorry, how do u divide everything by x?

I know 10x divide by x leaves 10.

I'm not sure how to divide the denominator by x, esp. when there is a square root over the x^2+10x. Thanks. If I'm plugging infinity into the equation, how is the answer 5?

 

[pka]

“Sorry, how do u divide everything by x?”
JohnJones, it appears that your problem is just as we suspected.
You have real issues with basic algebra. It is hard to do calculus without algebra.
If x>0, [√(x<SUP>2</SUP>+10x)+x]/x =√(1+(10/x)+1 .
If you do not see that, then if I were you, I would consider retaking algebra.

 

[tkhunny]

Really, you will have a great advantage over your fellow calculus students if you spend the year learning calculus and they are still learing algebra. It is an excellent way to distinguish yourself in an early calculus class.

While we're at it, how's your Analytic Geometry and Trigonometry? You'll need those, too. That's why they are called "prerequisites".