1) find the indicated limit, if it exist?

a) lim x->-2 (x^2 -9)/(x^2+x-2)
b) lim x -> -∞ √(ax^2+bx+c)/dx + e, where a > 0, b,c,d, and e are constant.

a)

since you have (x-3)(x+3) / (x-1)(x+2) the numerator is nonzero when x = -2, the limit will be ±∞, depending on the direction of approach,

b)
I assume you meant √(ax^2+bx+c)/(dx+e) since otherwise it is boring. Divide top and bottom by x to get
√(a+b/x+c/x^2)/(d+e/x) = √a/d
however, the numerator is positive and the denominator is negative as x -> -∞, so we really end up with -√a/d

To find the indicated limit, we need to evaluate the expression as x approaches the given value or infinity, and see if it approaches a specific value or goes to infinity. Let's calculate the limits for the provided expressions:

a) lim x -> -2 (x^2 - 9)/(x^2 + x - 2):

To find the limit as x approaches -2, we can directly substitute -2 into the expression and simplify:
lim x -> -2 (x^2 - 9)/(x^2 + x - 2) = (-2^2 - 9)/(-2^2 + (-2) - 2)
= (4 - 9)/(4 - 2 - 2)
= (-5)/0

We get an indeterminate form of (-5)/0, which indicates that the expression doesn't have a limit as x approaches -2. In this case, the limit does not exist.

b) lim x -> -∞ √(ax^2 + bx + c)/(dx + e), where a > 0, b, c, d, and e are constant:

To evaluate this limit, we need to consider the highest power term in the numerator and denominator. Since x approaches negative infinity, we focus on the term with the highest power, which is ax^2 in the numerator and dx in the denominator.

Let's divide both the numerator and denominator by x^2:
lim x -> -∞ √(ax^2/x^2 + bx/x^2 + c/x^2)/(dx/x^2 + e/x^2)
= lim x -> -∞ √(a + b/x + c/x^2)/(d/x + e/x^2)

As x goes to negative infinity, both (1/x) and (1/x^2) both approach 0. Therefore, we get:
lim x -> -∞ √(a + 0 + 0)/(0 + 0)
= √(a/0)

We get an indeterminate form of √(a/0), which indicates that the expression doesn't have a limit as x approaches negative infinity. In this case, the limit does not exist.

In summary, for both expressions (a) and (b), the limits do not exist.