Which combination of limit properties is required to evaluate this limit?

lim (24/x-2x+2)^3
x->4

f. sum, difference, product, root
g. sum, difference, product, power
h. sum, difference, quotient, power
i. sum, difference, quotient, root
j. A limit does not exist.

To evaluate the given limit, we can use a combination of limit properties. The properties we need are:

h. sum, difference, quotient, power.

Explanation:
1. Sum property: This property states that the limit of the sum of two functions is equal to the sum of their limits. In this case, we have the expression (24/x - 2x + 2)^3. We can break it down into three parts: 24/x, -2x, and 2.

2. Difference property: This property states that the limit of the difference of two functions is equal to the difference of their limits.

3. Quotient property: This property states that the limit of the quotient of two functions is equal to the quotient of their limits, provided the denominator doesn't approach zero.

4. Power property: This property states that the limit of a function raised to a power is equal to the limit of the function raised to that power.

By using the sum, difference, quotient, and power properties, we can evaluate the limit:

lim (24/x-2x+2)^3
x->4

= lim (24/x)^3 - (2x)^3 + 2^3
x->4

= lim (24^3/x^3) - (2x)^3 + 2^3
x->4

= 24^3/4^3 - 2^3*4^3 + 2^3
x->4

= 13824/64 - 8*64 + 8
x->4

= 216 - 512 + 8
x->4

= -288.

Therefore, the correct combination of limit properties required to evaluate this limit is option h. sum, difference, quotient, power.