Final Answer:
for x ≥0, the horizontal line y = 2 is an asymptote of the function f . The correct statement is lim x to 2 f(x) = infinity. Therefore, the correct option is D. lim x to 2 f(x) = infinity.
Step-by-step explanation:
In calculus, an asymptote is a line or curve that a function approaches but never touches as the input approaches a certain value. In this case, the function f has a horizontal asymptote at y = 2 for x ≥ 0. This means that as x approaches infinity or as x approaches 2 from the right, the value of f(x) approaches 2.
To determine which of the given statements is true, we can analyze each statement and see if it matches the definition of an asymptote.
A. f(0) = 2: This statement is false because an asymptote does not represent a value that the function takes on. Instead, it represents a value that the function approaches but never reaches.
B. f(x) not = 2 for all x greater than/= 0: This statement is also false because there may be values of x greater than or equal to 0 where f(x) = 2. The statement should instead say "f(x) does not equal 2 for all x greater than/equal to 0 except possibly at x = 2" to match the definition of an asymptote.
C. f(2) is undefined: This statement is irrelevant because it does not relate to the definition of an asymptote. An asymptote represents a value that the function approaches, not a value that the function takes on at a specific point.
D. lim x to 2 f(x) = infinity: This statement matches the definition of an asymptote because it says that as x approaches 2 from the right, the value of f(x) approaches infinity, which means that the graph of f gets infinitely close to the horizontal line y = 2 but never touches it.
E. lim x to infinity f(x) = 2: This statement is false because it suggests that as x approaches infinity, the value of f(x) approaches 2. However, we know from our analysis that as x approaches infinity, the value of f(x) approaches the horizontal line y = 2 but does not necessarily approach a specific value on that line. Therefore, this statement does not accurately describe an asymptote. Therefore, the correct option is D. lim x to 2 f(x) = infinity.