Question

find the horizontal asymptote f(x) = 2x + 1

Explanation:

Step1: Recall Horizontal Asymptote Rules

For a function \( f(x) \), horizontal asymptotes are determined by the end - behavior as \( x
ightarrow\pm\infty \). For a linear function of the form \( f(x)=mx + b \) (where \( m
eq0 \)):

• If \( m = 0 \), the horizontal asymptote is \( y = b \) (since the function is a constant function \( f(x)=b \)).
• If \( m

eq0 \), as \( x
ightarrow\pm\infty \), \( f(x)=mx + b \) will also tend to \( \pm\infty \) (depending on the sign of \( m \)). So, linear functions with non - zero slope do not have horizontal asymptotes.

Step2: Analyze the Given Function

The given function is \( f(x)=2x + 1 \), which is a linear function with slope \( m = 2
eq0 \) and \( y \) - intercept \( b = 1 \). As \( x
ightarrow\infty \), \( f(x)=2x + 1
ightarrow\infty \), and as \( x
ightarrow-\infty \), \( f(x)=2x + 1
ightarrow-\infty \).

Answer:

The function \( f(x) = 2x+1 \) has no horizontal asymptote.