limits

• Aug 31st 2008, 04:17 AM
hollies_yardbirds_kinks
limits
could someone please explain to me how to do these?

3x^2 + 2x + 1
limx --------------
x^2 + x - 3

x^2 + 1
limx ---------
x^2 - 1

thank you
• Aug 31st 2008, 04:18 AM
kalagota
Quote:

Originally Posted by hollies_yardbirds_kinks
could someone please explain to me how to do these?

3x^2 + 2x + 1
limx --------------
x^2 + x - 3

x^2 + 1
limx ---------
x^2 - 1

thank you

where do we approach x?
• Aug 31st 2008, 04:23 AM
Moo
Hello !
Quote:

Originally Posted by hollies_yardbirds_kinks
could someone please explain to me how to do these?

3x^2 + 2x + 1
limx --------------
x^2 + x - 3

x^2 + 1
limx ---------
x^2 - 1

thank you

When you have a division of polynomials (and limit to infinity), there are a few possible ways to solve the limit :
- l'hospital's rule
- find the greatest power of x in the numerator and the denominator and factor in the numerator and denominator. For example, if you have x²+1, factor by x² : $\displaystyle x^2+1=x^2 \left(1+\frac 1{x^2}\right)$. $\displaystyle x^2-1=x^2 \left(1-\frac 1{x^2}\right)$
So you can simplify by x². What's left is in general a finite limit

- know some rules : if the degree of the polynomial in the numerator is greater than the one of the denominator, the limit is $\displaystyle \infty$ (the sign depends on the leading coefficient)
if the degree of the polynomial in the numerator is smaller than the one of the denominator, the limit is 0.
if they're the same, the limit is the quotient of the leading coefficients.
These rules derive from the point just above :)

Sidenote : Leading coefficient : Definition