Evaluate this limit, introductory real analysis

[jaqueh]

Homework Statement

limit of the sequence, [xn]=(-3n²+n+1)/(n²-2n+3)

Homework Equations

I so far know about the definition of a limit, squeeze principle, and lim[xnyn] = 0 if xn or yn goes to 0

The Attempt at a Solution

Tried the definition of the limit but the algebra got really crazy so i don't think I'm supposed to do it that way. I'm trying to squeeze it between (-3n²/n²) and (-3n³)/(n³), but i don't know if i can prove that the cubed one is greater than my sequence

 

[scurty]

Try factoring out a variable of the highest power. This is a standard method for problems like these before l'Hospital's rule is introduced.

Edit: Just in case you aren't allowed to use that method..

If you want to use the squeeze theorem and then the definition of the limit, you need to factor out some terms by manipulating the original expression so they cancel out.

 

[jaqueh]

ah I've divided the numerator by 3 but am still stuck. I know it hits -3.

yeah I'm trying to see what I can make the expression into so things will start cancellong

 

[HallsofIvy]

Did you misunderstand what scurty said? Nothing about dividing by -3: factor out, or, same thing, divide both numerator and denominator by $n^2$.

 

[jaqueh]

ok great I have it now. I do that then squeeze because I'm not allowed to say yet that 1/n goes to zero in a sequence I think