If lim f(x)-3/x-1 =2 Find lim f(x) x--->1 x--->1 Solution If lim(x->3) { [f(x) -

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If lim(x->3) { [f(x) - 5]/(x-3) } = 4 Then the numerator must have two factor, and one of them is (x-3) because you need to cancel it with the denominator (x-3) in order to have the function defined. After the two (x-3)'s are canceled, you are left with the other factor. And since you are suppose to substitute in the value a = 3, the only way to get 4 as an answer is to have the other factor as (x+1). So the numerator was originally (x-3)(x+1). However, if expanded, this would equal x^2 -2x - 3. But in order to have -5 at the end, fix -3 into two parts that include -5. For example, -3 = -5 + 2 Therefore, f(x) = x^2 - 2x + 2 If you want to check, go ahead. lim(x->3) [f(x)-5]/(x-3) = lim(x->3) [(x^2 - 2x + 2) - 5]/(x-3) = lim(x->3) [x^2 - 2x - 3] / (x-3) = lim(x->3) [(x-3)(x+1)] / (x-3) = lim(x->3) (x+1) = 3 + 1 = 4

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