Limit Evaluation Help: How to Simplify a Limit Involving Rational Functions

[f(x)]

Homework Statement

Evaluate: $$\lim_{x \rightarrow \infty} ( \frac {x^3}{(3x)^2-4}-\frac{x^2}{3x+2} )$$2. The attempt at a solution

First i took x^2/3x+2 common, and then substituted y=1/x , y->0.
simplifying, i get $$\frac{-2}{9y}$$...i am unable to eliminate all the y's.
The answer given at the back of the book is 2/9.
Plz help me figure this out.
&&Thx

 

[learningphysics]

Are you sure that's exactly the question? The limit approaches minus infinity...

 

[P.O.L.A.R]

There is a theorem for this because what you have is an $$\infty$$/$$\infty$$ correct? (L'Hopital)
Note: If this is a calc one qustion ignore this.

 

[f(x)]

Are you sure that's exactly the question? The limit approaches minus infinity...

yeah, that's why i am confused

The trick is to convert each term with x's into a term with 1/x's.

What do you need to divide with both the numerator and the denominator of the first term, to convert all the x's into 1/x's?

What do you need to divide with both the numerator and the denominator of the second term?

Say for the 1st term, i divide and multiply by x^3, the Dr. becomes 0 ?

 

[learningphysics]

Are you sure the question isn't:

$$\lim_{x \rightarrow \infty} ( \frac {3x^3}{(3x)^2-4}-\frac{x^2}{3x+2} )$$

 

[rock.freak667]

i keep getting $$\infty$$ not 2/9

 

[d_leet]

Find a common denominator, then add the fractions together.

 

[f(x)]

Are you sure the question isn't:

$$\lim_{x \rightarrow \infty} ( \frac {3x^3}{(3x)^2-4}-\frac{x^2}{3x+2} )$$

I checked it again, but the question is exactly as I put it.

 

[malawi_glenn]

(3x)^2 - 4 = 9x^2 - 4 = (3x + 2)(3x - 2)

use this to simply your function (i.e multply your second term with (3x -2) in both nominator and denominator)

Then the x^3 term will vanish

Also, there must be a misprint as "learningphysics" has noticed

 

[learningphysics]

I checked it again, but the question is exactly as I put it.

I'm guessing it's a misprint or something that left out the 3... because the limit of the function I just posted is 2/9.