Question

question 1 (1 point)
a function f(x) has a horizontal asymptote at y = l if (select all that apply):
□ $lim_{x
ightarrow-infty}f(x)=l$
□ $lim_{x
ightarrowinfty}f(x)=l$
□ $lim_{x
ightarrow l}f(x)=-infty$
□ $lim_{x
ightarrow l}f(x)=infty$

Explanation:

Step1: Recall horizontal - asymptote definition

A horizontal asymptote of a function \(y = f(x)\) is a horizontal line \(y = L\) such that the function approaches \(L\) as \(x\) approaches positive or negative infinity.

Step2: Analyze each option

The limit \(\lim_{x
ightarrow-\infty}f(x)=L\) means the function approaches \(L\) as \(x\) goes to negative infinity, which indicates a horizontal asymptote at \(y = L\). The limit \(\lim_{x
ightarrow\infty}f(x)=L\) means the function approaches \(L\) as \(x\) goes to positive infinity, which also indicates a horizontal asymptote at \(y = L\). The limits \(\lim_{x
ightarrow L}f(x)=-\infty\) and \(\lim_{x
ightarrow L}f(x)=\infty\) describe vertical - asymptote behavior (where the function goes to infinity as \(x\) approaches a particular value \(L\)), not horizontal - asymptote behavior.

Answer:

\(\lim_{x
ightarrow-\infty}f(x)=L\), \(\lim_{x
ightarrow\infty}f(x)=L\)