for x > 0, the horizontal line y = 2 is an asymptote for the graph of the function f. which of the following…

Explanation:

Step1: Recall definition of horizontal asymptote

A horizontal line $y = L$ is a horizontal asymptote of the function $y = f(x)$ if either $\lim_{x\rightarrow\infty}f(x)=L$ or $\lim_{x\rightarrow-\infty}f(x)=L$. Given that for $x > 0$, $y = 2$ is a horizontal asymptote, we know that $\lim_{x\rightarrow\infty}f(x)=2$.

Step2: Analyze each option

• Option a: Just because $\lim_{x\rightarrow\infty}f(x) = 2$ for $x>0$, we cannot say $f(0)=2$. The value of the function at a particular point $x = 0$ has no relation to the horizontal - asymptote behavior as $x\rightarrow\infty$.
• Option b: The fact that $y = 2$ is a horizontal asymptote for $x>0$ does not tell us anything about the value of $f(2)$. The function could be defined or undefined at $x = 2$ and it has no bearing on the horizontal - asymptote.
• Option c: $\lim_{x\rightarrow2}f(x)=\infty$ is not related to the fact that $y = 2$ is a horizontal asymptote for $x>0$. The limit as $x\rightarrow2$ is about the behavior of the function near $x = 2$, not as $x\rightarrow\infty$.
• Option d: Since $y = 2$ is a horizontal asymptote for $x>0$, by the definition of a horizontal asymptote, $\lim_{x\rightarrow\infty}f(x)=2$.

Answer:

d. $\lim_{x\rightarrow\infty}f(x)=2$