Let g and h be nonnegative functions with limits 1 and infinity, respectively, as x → a. What is the limit of the function below as x → a? If it cannot be determined, write the indeterminate form.

(9 + g(x))^−h(x)

I don't understand, please explain!

1/ [9 +1]^oo

1/10^oo

0

thanks!

will you please elaborate???

To find the limit of the given function as x approaches a, we need to evaluate the limit of the function when x approaches a. However, to determine the limit, we need to consider the behavior of the functions g(x) and h(x) as x approaches a.

Given that g(x) has a limit of 1 as x approaches a, we can say that g(x) approaches 1 as x approaches a.

Similarly, h(x) has a limit of infinity as x approaches a. This means that h(x) becomes arbitrarily large as x approaches a.

Now, let's analyze the function (9 + g(x))^−h(x) as x approaches a. Since g(x) approaches 1 and h(x) approaches infinity, we can rewrite the function as (9 + 1)^−∞.

When we evaluate (9 + 1)^−∞, we get an indeterminate form. An indeterminate form is a limit that cannot be determined just by observing the functions involved. In this case, the indeterminate form is 1^−∞.

To further evaluate this indeterminate form, we can use a limit property. The property states that if f(x) → c and g(x) → ∞ as x approaches a, then (f(x))^g(x) approaches either 0 or infinity, depending on the value of c.

In our case, (9 + 1)^−∞ can be rewritten as 10^−∞. Since 10 is a constant and not approaching 0 or infinity, the limit of this function cannot be determined using this property alone.

Therefore, the limit of the function (9 + g(x))^−h(x) as x approaches a is indeterminate.