(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).
I don't understand the = sign. Limit is an expression, not an equation.
To find the number c such that the limit in part (a) exists, we need to evaluate the limit for different values of c and see if it approaches a finite number as x approaches -2.
(a)
To evaluate the limit, we substitute x = -2 into the expression and see if it approaches a finite number as c changes.
Let's substitute x = -2 into the expression:
Limit as x approaches -2 of: x^2 + cx / (c - 3 / (x^2 + 2x))
Plugging in x = -2, we have:
(x^2 + cx) / (c - 3 / (x^2 + 2x))
= (-2)^2 + c(-2) / (c - 3 / (-2)^2 + 2(-2))
= 4 - 2c / (c - 3 / 4 - 4)
Simplifying further, we have:
= 4 - 2c / (c - 3 / 4 + 8)
= 4 - 2c / (c - 3 + 32)
= 4 - 2c / (c + 29)
Now, we need to determine the value of c that makes the limit exist. For the limit to exist, we need the denominator (c + 29) to be non-zero. Therefore, c cannot be equal to -29.
So, any value of c except c = -29 will make the limit in part (a) exist.
(b)
To calculate the limit for the value of c in part (a), we substitute c = -29 into the expression and evaluate the limit:
Limit as x approaches -2 of: x^2 + cx / (c - 3 / (x^2 + 2x))
= 4 - 2(-29) / (-29 + 29)
= 4 + 58 / 0
The denominator is 0, which means the limit is undefined for c = -29.
In conclusion, the limit in part (a) exists for any value of c except c = -29, and the limit is undefined for c = -29.