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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:
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
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.
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
so option B is the right answer
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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