The function [tex]$f(x)=\frac{1}{x+3}$[/tex] has a horizontal asymptote at
A. [tex]$x=0$[/tex]
B. [tex]$f(x)=0$[/tex]
C. [tex]$f(x)=-3$[/tex]
D. [tex]$f(x)=3$[/tex]
Answer (2)
The function f ( x ) = x + 3 1 has a horizontal asymptote at f ( x ) = 0 as both limits approach 0 when x approaches infinity and negative infinity. Therefore, the correct answer is B. f ( x ) = 0 .
;
Find the limit of f ( x ) = x + 3 1 as x approaches infinity.
Find the limit of f ( x ) = x + 3 1 as x approaches negative infinity.
Both limits are 0.
The horizontal asymptote is f ( x ) = 0 .
Explanation
Understanding Horizontal Asymptotes The problem asks us to find the horizontal asymptote of the function f ( x ) = x + 3 1 . A horizontal asymptote is the value that the function approaches as x goes to infinity or negative infinity.
Limit as x approaches infinity To find the horizontal asymptote, we need to calculate the limit of the function as x approaches infinity and negative infinity. Let's start with the limit as x approaches infinity: x → ∞ lim x + 3 1
Evaluating the Limit at Infinity As x becomes very large, the term x + 3 also becomes very large. Therefore, the fraction x + 3 1 approaches 0. So, x → ∞ lim x + 3 1 = 0
Limit as x approaches negative infinity Now, let's consider the limit as x approaches negative infinity: x → − ∞ lim x + 3 1
Evaluating the Limit at Negative Infinity As x becomes a very large negative number, the term x + 3 also becomes a very large negative number. Therefore, the fraction x + 3 1 approaches 0. So, x → − ∞ lim x + 3 1 = 0
Conclusion Since the function approaches 0 as x approaches both infinity and negative infinity, the horizontal asymptote is f ( x ) = 0 .
Examples
Understanding horizontal asymptotes is crucial in various real-world applications. For example, in pharmacology, the concentration of a drug in the bloodstream over time can be modeled by a function with a horizontal asymptote. The asymptote represents the steady-state concentration of the drug, which is essential for determining appropriate dosages. Similarly, in environmental science, the long-term concentration of a pollutant in a lake might be modeled with a function that has a horizontal asymptote, indicating the eventual equilibrium level of pollution.
