Suppose that

h(x)={x^2-x+5 if x<2
{5 if x=2
{x^3-1 if x>2
Which of the following is equal to 7?
I. lim h(x)
x->2-
II. lim h(x)
x->2+
III. lim h(x)
x->2

I only
II only
***III only
I and II only
I, II, and III

Well, to find the values of the limits, we need to evaluate the function h(x) for each case. Let's start with lim h(x) as x approaches 2 from the left, which is represented as lim h(x) x->2-. In this case, x is smaller than 2, so we use the first part of the function, h(x) = x^2 - x + 5.

Plugging in x = 2 into the first part of the function, we get: h(2) = (2^2) - 2 + 5 = 4 - 2 + 5 = 7.

Next, let's look at lim h(x) as x approaches 2 from the right, which is represented as lim h(x) x->2+. In this case, x is greater than 2, so we use the third part of the function, h(x) = x^3 - 1.

Plugging in x = 2 into the third part of the function, we get: h(2) = (2^3) - 1 = 8 - 1 = 7.

Finally, let's consider lim h(x) as x approaches 2, which is represented as lim h(x) x->2. In this case, since the function is defined for x=2, we use the second part of the function, h(x) = 5.

Therefore, we have: lim h(x) x->2- = lim h(x) x->2+ = lim h(x) x->2 = 7.

So, the correct answer is III only.

To find which of the following limits is equal to 7, we need to evaluate each limit:

I. lim h(x) as x approaches 2 from the left (x->2-):
We take the limit as x approaches 2 from the left which means x is getting closer and closer to 2, but is always less than 2. From the definition of h(x), we have h(x) = x^2 - x + 5 if x < 2. Plugging in x = 2, we get h(2) = 2^2 - 2 + 5 = 7. Thus, the limit as x approaches 2 from the left equals 7.

II. lim h(x) as x approaches 2 from the right (x->2+):
We take the limit as x approaches 2 from the right which means x is getting closer and closer to 2, but is always greater than 2. From the definition of h(x), we have h(x) = x^3 - 1 if x > 2. Plugging in x = 2, we get h(2) = 2^3 - 1 = 7. Thus, the limit as x approaches 2 from the right equals 7.

III. lim h(x) as x approaches 2 (x->2):
To find this limit, we need to consider both the left and right limits. From the previous calculations, we found that the left limit and right limit are both equal to 7. Since the left and right limits are equal, the limit as x approaches 2 is also equal to 7.

Therefore, the correct answer is III only.

To determine which of the given options is equal to 7, we need to evaluate the limit of h(x) as x approaches 2 from the left (lim h(x) as x->2-) and the limit of h(x) as x approaches 2 from the right (lim h(x) as x->2+), as well as the limit of h(x) as x approaches 2 (lim h(x) as x->2).

I. lim h(x) as x approaches 2 from the left (lim h(x) as x->2-):
To evaluate this limit, we substitute values of x that are slightly less than 2 into the function h(x).

Let's substitute x = 1.9:
h(1.9) = (1.9^2 - 1.9 + 5) = 7.41

Let's substitute x = 1.99:
h(1.99) = (1.99^2 - 1.99 + 5) = 7.0399

As x approaches 2 from the left, the values of h(x) are approaching 7.

II. lim h(x) as x approaches 2 from the right (lim h(x) as x->2+):
To evaluate this limit, we substitute values of x that are slightly greater than 2 into the function h(x).

Let's substitute x = 2.1:
h(2.1) = (2.1^3 - 1) = 9.261

Let's substitute x = 2.01:
h(2.01) = (2.01^3 - 1) = 9.0601001

As x approaches 2 from the right, the values of h(x) are approaching 9.

III. lim h(x) as x approaches 2 (lim h(x) as x->2):
To evaluate this limit, we consider both the left and right limits of h(x) as x approaches 2.

Since the left limit is approaching 7 and the right limit is approaching 9, the limit of h(x) as x approaches 2 does not exist.

Therefore, the option that is equal to 7 is III only.

Final Answer: III only

looks good.