Question

Given that lim x→a f(x) = 0, lim x→a g(x) = 0, lim x→a h(x) = 1, lim x→a p(x) = [infinity], lim x→a q(x) = [infinity], evaluate if the following limits are not indeterminate forms. (If a limit is an indeterminate form, write 'indeterminate'.)

1) 0
2) 1
3) [infinity]
4) indeterminate

Answer 1

The given limits are 0, 1, infinity, and 'indeterminate' respectively, and with the exception of the last which is simply described as 'indeterminate,' none of the other limits provided are considered indeterminate forms.

Step-by-step explanation:

When evaluating if the following limits are not indeterminate forms, we must consider each separately:

1. The limit of f(x) as x approaches a is given to be 0. Therefore, this limit is simply 0, which is not an indeterminate form.
2. Similarly, the limit of h(x) as x approaches a is 1. This limit is also not an indeterminate form.
3. The limit of p(x) as x approaches a becomes infinity, which is not considered an indeterminate form but does indicate that the function increases without bound.
4. However, if we encounter expressions such as 0/0 or infinity/infinity, these would be considered indeterminate forms, but with the information given, this limit is simply described as 'indeterminate.'

For each scenario, the specific values of the limits provided do not fall under the classical indeterminate forms such as 0/0, ∞/∞, 0∗∞, ∞−∞, 1^∞, 0^0, and ∞^0.