Hi skeske1234
If you know 3 is the facor of the numerator, can you factorize (x^3+x^2-8x-12) ?
Hi, I'm not sure how to evaluate the following limits.
1. lim (x^3+x^2-8x-12)/(x+2)
x->-2
so I know that I have to use the factor theorem for the numerator, and I noticed that 3 is a factor of the numerator. so I'm not sure how to evalute this limit or proceed to the next step with this in mind (using the factor theorem).
2. lim(27-x)/(x^(1/3)-3)
x->27
for this one, i am not sure if i have to change the variables and if so, what i have to change it to??
Please explain the steps needed. Thank you in advance
The zeros of the numerator are not of primary importance here. The problem is that the denominator has a zero at the very point where we want to take the limit! So if you first check (by polynomial division) whether the factor occurs in the numerator you find that
Remember ? Now substitute for (and therefore for ), and you find that2. lim(27-x)/(x^(1/3)-3)
x->27
for this one, i am not sure if i have to change the variables and if so, what i have to change it to??
Please explain the steps needed. Thank you in advance
Do you mean that "3" is a factor of the numerator or that "x- 3" is a factor?
In any case, that is not really relevant. What is very relevant is that x+2, the denominator, goes to 0 as x goes to -2 so "x+2" had better be a factor! You can show that it is, and find the factorization by dividing by x+ 2, either by direct division or by "synthetic" division. You could also "work" it out by thinking "if x+2 is a factor, then it must be something like Multiply out the left side and see what "?" must be.
(If you know x+ 2 must be a factor and you also know that x- 3 is a factor, then it is easy to see that [tex]x^3+ x^2- 8x- 12= (x+ 2)(x- 3)(x- ?) and since -12= 2(-3)(2), what is "?" ?
Again, you need to be able to cancel that in the denominator. Fortunately, we can precisely because 27 also makes the numerator 0. Remember that with and .2. lim(27-x)/(x^(1/3)-3)
x->27
for this one, i am not sure if i have to change the variables and if so, what i have to change it to??
By the way, you might want to start posting in "Calculus". You may be taking a "Pre-Calculus" but these are definitely introductory Calculus questions.Please explain the steps needed. Thank you in advance