Question

Is there a number a such that the following limit exists? (If an answer does not exist, enter DNE.) lim x→-2 (2x² + ax)/(a⁶x²(x - 2)) Find the value a.

Answer 1

The value of a that makes the given limit exist is 0.

Step-by-step explanation:

The limit of the given expression can be found by evaluating the expression for x approaching -2. To find the value of a, we can substitute x = -2 into the expression and solve for a. Let's calculate:

limx→-2 (2x² + ax)/(a⁶x²(x - 2))

Substituting x = -2:

(2(-2)² + a(-2))/(a⁶(-2)²(-2 - 2))

(8 - 2a)/(16a⁶)

For the limit to exist, the denominator must not be zero. Therefore, we set the denominator equal to zero and solve for a:

16a⁶ = 0

a⁶ = 0

a = 0