Find the constant c so that lim x->8 x^2+2x+c/x^2-15x+56 exists. For that value of c, determine the limit. (Hint: Find the value of c for which x-8 is a factor of the numerator).
I understand that I should set (x-8)(x+b) equal to x^2+2x+c but all the variables are getting confusing.
(x-8)(x+b) = x^2 + 2 x + c
- 8 b = c
and
-8 x + b x = 2 x
or
b = 10
so
c = -80
To solve this problem, you are correct in setting up the equation (x-8)(x+b) = x^2+2x+c, where b represents the other factor of the numerator. However, to simplify the problem, you can use the concept of the Remainder Theorem.
The Remainder Theorem states that if a polynomial f(x) is divided by (x-a), the remainder will be zero if and only if f(a) = 0. In this case, you want (x-8) to be a factor of the numerator, which means that when x = 8, the numerator should be zero.
Let's substitute x = 8 into the equation: (8^2+2*8+c)/(8^2-15*8+56) = 0.
Simplifying this equation, we get: (64 + 16 + c)/(64 - 120 + 56) = 0.
Further simplifying, we have: (80 + c)/(0) = 0.
Since the denominator is zero (which means the function is undefined), the numerator must also be zero for the limit to exist.
Therefore, we can conclude that 80 + c = 0. Solving for c, we get c = -80.
So the constant c should be -80 for the limit to exist. To determine the value of the limit, substitute this value into the expression: lim x->8 (x^2+2x-80)/(x^2-15x+56).
I hope this explanation helps! Let me know if you have any further questions.