(a) Find the number c such that the limit below exists.
Limit as x goes to -2 of: x^2+cx=c-3/x^2+2x
(b) Calculate the limit for the value of c in part (a).
To find the value of c such that the limit exists, we need to evaluate the limit as x approaches -2 for the given expression and then find the value of c that satisfies this limit.
(a) Let's calculate the limit as x approaches -2 of the expression:
Limit as x goes to -2 of: (x^2 + cx) / (c - 3/x^2 + 2x)
To evaluate this limit, we substitute -2 for x in the expression:
(x^2 + cx) / (c - 3/x^2 + 2x) = (-2^2 + c(-2)) / (c - 3/(-2)^2 + 2(-2))
= (4 - 2c) / (c - 3/4 - 4)
= (4 - 2c) / (c - 3/4 - 16/4)
= (4 - 2c) / (c - 19/4)
In order for the limit to exist, the denominator must not be approaching zero as x approaches -2. Therefore, we set the denominator equal to zero and solve for c:
c - 19/4 = 0
c = 19/4
Therefore, the number c that makes the limit exist is c = 19/4.
(b) Now, let's calculate the limit for the value of c in part (a). Substituting c = 19/4 into the expression:
Limit as x goes to -2 of: (x^2 + (19/4)x) / (19/4 - 3/x^2 + 2x)
Again, substitute -2 for x in the expression:
((-2)^2 + (19/4)(-2)) / (19/4 - 3/(-2)^2 + 2(-2))
= (4 - 19/2) / (19/4 - 3/4 - 8/4)
= (-11/2) / (19/4 - 11/4)
= (-11/2) / (8/4)
= (-11/2) * (4/8)
= -11/4
Therefore, the limit for the value of c in part (a) is -11/4.