How to do this limit

Question

lim as x tends to -1+ of (x^2-2x+1)/(x+1) 

 

I did long division amd got (x-3) + (4)/(x+1) then I just plugged -1 in and got -4 but the solutions  manual is says otherwise. 

Answer 1

Alex,

 

That can be factored and rewritten as (x-1)²/(x+1).

 

Since you'd be dividing by zero, there is an asymptote at x=-1. We need to find the value that is just to the right of x=-1.

 

Let's think about the denominator. If we plug in x=-0.9, we get 0.1 in the denominator. Having a small number in the denominator means that your function typically will be large.

 

Solve for x=0.9. We get (-1.9)²/(0.1). This equals 36.1.

 

The closer we get to x=-1 on the right side, the larger value we will get. I we plug in x=0.9999, you would get an incredibly tiny number on the denominator.

 

This being said, your limit x -> -1⁺ is going to equal +∞.