Question

Match the expression for the limit of the function with the function's horizontal asymptote if it has one.

a. lim→[infinity]()lim x→[infinity] f(x)
b. lim→−[infinity]()lim x→−[infinity]f(x)
c. lim→[infinity]()−lim→−[infinity]()lim x→[infinity] f(x)−lim x→−[infinity] f(x)
Match each expression with the corresponding description of the function's horizontal asymptote.
1. No horizontal asymptote
2. y=0=
3. y=c (where c is a constant)
a. Expression a matches with ____.
b. Expression b matches with ____.
c. Expression c matches with ____.

Answer 1

Expression a and b match with a horizontal asymptote at y=c where c is a constant, and expression c matches with no horizontal asymptote since it is a difference that does not define a horizontal level.

Step-by-step explanation:

To match the expressions for the limit of the function with the function's horizontal asymptote, consider the following:

Expression a, lim x→∞ f(x), corresponds to the value that f(x) approaches as x goes to infinity. If f(x) approaches a constant value c, then the function has a horizontal asymptote at y = c. If it does not approach a constant value, then there is no horizontal asymptote.

Similarly, expression b, lim x→-∞ f(x), corresponds to the function's behavior as x approaches negative infinity.

Expression c, lim x→∞ f(x) − lim x→-∞ f(x), does not directly represent a horizontal asymptote, as it is the difference between the limits as x tends to infinity and negative infinity. If this difference is finite, the function might have two horizontal asymptotes at different levels, or none if the difference is not constant.
Based on the information:

Expression a matches with 3. y = c (where c is a constant).

Expression b matches with 3. y = c (where c is a constant).

Expression c matches with 1. No horizontal asymptote.