What is the algebraic approach to finding limits approaching infinity?

[1MileCrash]

I'm having a hard time learning from the textbook, I know I can do this if someone just outlines what my goal is here... and what I can interpret from that goal.

The solutions handbook just makes seemingly random algebraic changes to the limit function and then tells me what the answer is, and it's a bit confusing.

Homework Statement

Find the limit approaching infinity of:

[3sqrt(x^3)] / [sqrt(2x^3)]

Homework Equations

The Attempt at a Solution

I don't know where to begin. Don't solve it for me, explain what I am looking to get through the algebraic changes, and I will post an attempted solution.

Thanks.

 

[cragar]

what would be the limit of $\frac{2x^2}{x^2}$
as it went to infinity

 

[LCKurtz]

It helps to state the problem clearly. You want the limit as x approaches infinity. Your first step should be to simplify the algebraic expression you are working with.

 

[1MileCrash]

Alright, after looking through more examples it looks like they want me to try to get to n/x (x approaching infinity) which becomes zero.

I also wrote the problem incorrectly, 3x should be added to the numerator. So will post back with an attempt shortly.

 

[Mentallic]

Alright, after looking through more examples it looks like they want me to try to get to n/x (x approaching infinity) which becomes zero.

I also wrote the problem incorrectly, 3x should be added to the numerator. So will post back with an attempt shortly.

Let's get this straight, the question is to find

$$\lim_{x\to \infty}\frac{3x+3\sqrt{x^3}}{\sqrt{2x^3}}$$

?

 

[1MileCrash]

Yes, but I completely understand my goal now. I'm just dividing every term by the highest x power in the denominator and it's working out eventually as n/x (0). My answer is 3/sqrt(2).

 

[HallsofIvy]

$$\sqrt{x^3}= x^{3/2}$$
so this is
$$\frac{3x+3x^{3/2}}{\sqrt{2}X^{3/2}}= \frac{3}{\sqrt{2}}\frac{x+ x^{3/2}}{x^{3/2}}$$

The highest power of x is 3/2 so divide both numerator and denominator by $x^{3/2}$ to get
$$\frac{3}{\sqrt{2}}\frac{x^{-1/2}+ 1}{1}$$

That gives exactly what you say. Good work!