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gsenviro eNotes educator| Certified Educator

Direct substitution will lead to a 0/0 situation (you can check it yourself), hence we can find a common factor int he numerator and denominator and cancel it out and use Dividing out technique to solve it.

The numerator can be simplified as:

`1-2x-3x^2 = -(3x^2 +2x-1) = -(3x-1)(x+1)`

and we can cancel out this common factor (x+1) from numerator and denominator.

And we will be left with only -(3x-1) in the numerator.

therefore, `lim_(x->-1) (1-2x-3x^2)/(1+x) = lim_(x->-1)-(3x-1) `

`= -(3*(-1) -1) = 4`

Thus, we can use dividing out technique (option #2) to get the answer as 4.

Hope this helps.

steveschoen eNotes educator| Certified Educator

Many times, with ones like these, the top can be factored into two binomials, one of them being the bottom.  In this case, it can.  So, we would have:

`((1+x)(1-3x))/((1+x))`

So, the "1+x" would cancel out.  To then evaluate the limit, plug in -1 for x:

1 - 3(-1) = 1 - (-3) = 1 + 3 = 4

kspcr111 | Student

Given to solve :

`lim_(x-> -1) ((1-2x-3x^2)/(1+x))`

 `=lim_(x-> -1) ((1+x-3x-3x^2)/(1+x))`

 = `lim_(x-> -1) ((1+x)-3x(1+x))/(1+x))` [ this is Dividing out technique]

  = `lim_(x-> -1) ((1+x)(1-3x))/(1+x))`

  =`lim_(x-> -1) (1-3x)`

  applying the values of `x= -1` , we get 

  = `1-3(-1)`

   = `4`

so option B is the right answer 

shmindle | Student

You would choose the 2nd option - the dividing out technique.

You can simplify the numerator to: (-3x+1)(x+1)

Then, the (x+1) from the numerator and denominator would cancel out, leaving you with (-3x+1).

Plug -1 into (-3x+1).  you will get 4, which is the limit of the graph as x approaches -1. 

You can use your graphing calculator to verify.  

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