State which of the conditions are applicable to the graph of
y = f(x).
(Select all that apply.)
lim x→infinity f(x) = −infinity
lim x→a+ f(x) = L
lim x→infinity f(x) = L
f is continuous on [0, a]
lim x→infinity f(x) = 0
f(a) is not defined
lim x→a |f(x)| = infinity
f(a) = L
f is continuous at x = a
lim x→a f(x) = L
All of the above are true. Isn't f(x) a marvelous function?
To determine which conditions are applicable to the graph of y = f(x), we need to understand the meaning of each condition.
1. lim x→infinity f(x) = −infinity:
This condition means that as x approaches infinity, the value of f(x) approaches negative infinity. To check if this condition is applicable, we would need to evaluate the limit of f(x) as x tends to infinity. This can be done by analyzing the behavior of the function as x gets larger and larger.
2. lim x→a+ f(x) = L:
This condition refers to the limit of f(x) as x approaches a from the right side. If the function f(x) has a defined limit as x approaches a from the positive side, and the limit is L, then this condition is applicable.
3. lim x→infinity f(x) = L:
This condition states that as x approaches infinity, the value of f(x) approaches a specific finite value L. To verify if this applies, we would need to find the limit of f(x) as x goes to infinity.
4. f is continuous on [0, a]:
This condition indicates that the function f(x) is continuous for all x values in the closed interval [0, a]. Continuity means that there are no sudden jumps or discontinuities in the graph.
5. lim x→infinity f(x) = 0:
This condition says that as x approaches infinity, the value of f(x) tends to zero. We would need to evaluate the limit of f(x) as x approaches infinity to check if this condition holds.
6. f(a) is not defined:
This condition implies that the function f(x) does not have a defined value at x = a. This might indicate a discontinuity or a hole in the graph.
7. lim x→a |f(x)| = infinity:
This condition states that as x approaches a, the absolute value of f(x) tends to infinity. We would need to find the limit of the absolute value of f(x) as x approaches a to determine if this condition applies.
8. f(a) = L:
This condition means that the value of f(x) at x = a is equal to L. It implies that the function has a specific value at that point.
9. f is continuous at x = a:
This condition indicates that the function f(x) is continuous at x = a. This means there are no jumps, holes, or discontinuities at that specific point.
10. lim x→a f(x) = L:
This condition describes the limit of f(x) as x approaches a. If the limit of f(x) as x tends to a is equal to L, then this condition holds.
Based on these explanations, you can now determine which conditions are applicable to the graph of y = f(x) by applying the respective definitions and evaluating the limits and values of f(x) accordingly.