How Do You Calculate the Limit of a Function as x Approaches Negative Infinity?

[thushanthan]

Homework Statement

Find the limit of f(x) as x-> - (Infinity)

Homework Equations

(x⁴+1)(3-2x)³/(x+1)⁵(2-x²)

The Attempt at a Solution

I knew that for rational functions we have to divide both numerator and denominator by the highest power of x in the denominator, but this is confusing. Should I expand the equations? Please help ??

 

[bur7ama1989]

I'm not sure, but I think L'hopital's rule may work.

http://en.wikipedia.org/wiki/L'Hôpital's_rule

 

[thushanthan]

I am not allowed to use L'Hopital Rule

 

[elect_eng]

... Should I expand the equations? Please help ??

That is probably the most direct way to get the answer. Just expand the polynomials out in the numerator and the denominator.

lim_{x \to \infty} \frac{8\,{x}^{7}-36\,{x}^{6}+54\,{x}^{5}-27\,{x}^{4}+8\,{x}^{3}-36\,{x}^{2}+54\,x-27}{{x}^{7}+5\,{x}^{6}+8\,{x}^{5}-15\,{x}^{3}-19\,{x}^{2}-10\,x-2}

Then, as x goes to infinity, only the highest order terms matter.

=lim_{x \to \infty} \frac{8\,{x}^{7}}{{x}^{7}}

From here, the answer of 8 should be clear.

 

[Dick]

Your original plan looked ok to me. Imagine multiplying numerator and denominator out, without actually doing it. The highest power will be x^7, right? So divide by x^7. In the numerator split x^7=x^4*x^3. Divide the first factor by the x^4 and the second by x^3. In the denominator split it into x^7=x^5*x^2.

 

[thushanthan]

Thank you

Since this question is based on my calculus exam, I think I won't find time to multiply the whole polynomial. Therefore I will try to split and divide :)

 

[elect_eng]

Thank you

Since this question is based on my calculus exam, I think I won't find time to multiply the whole polynomial. Therefore I will try to split and divide :)

That is a good idea. Dick always gives good advice.

I'd like to point out that, in my suggestion, I was hoping that you would see that once you solve one of these problems, you really don't need to multiply out the whole polynomial. Only the highest order terms matter here. One can get to the step =lim_{x \to \infty} \frac{-8\,{x}^{7}}{{-x}^{7}} by multiplying out the highest order terms in your head.