Answer :
To determine the horizontal asymptote of the function \( f(x) = \frac{1}{x + 3} \), we need to evaluate the behavior of the function as \( x \) approaches infinity or negative infinity.
### Step-by-Step Solution:
1. Understand Horizontal Asymptotes:
- Horizontal asymptotes are lines that the graph of a function approaches as \( x \) heads towards positive or negative infinity.
- For rational functions, the horizontal asymptote can often be found by examining the degrees of the numerator and the denominator.
2. Evaluate the Function as \( x \to \infty \):
- Consider \( f(x) = \frac{1}{x + 3} \).
- To find the horizontal asymptote, we evaluate the limit of \( f(x) \) as \( x \) approaches infinity:
[tex]\[ \lim_{x \to \infty} \frac{1}{x + 3} \][/tex]
- As \( x \) becomes very large, the value of \( x + 3 \) also becomes very large. Thus, the term \(\frac{1}{x + 3}\) approaches 0.
3. Mathematical Result:
- So:
[tex]\[ \lim_{x \to \infty} \frac{1}{x + 3} = 0 \][/tex]
- This means that as \( x \) approaches infinity, the value of the function \( f(x) \) gets closer and closer to 0.
4. Conclusion:
- The horizontal asymptote of the function \( f(x) = \frac{1}{x + 3} \) is \( y = 0 \).
Thus, the correct answer is:
D. [tex]\( f(x) = 0 \)[/tex]
### Step-by-Step Solution:
1. Understand Horizontal Asymptotes:
- Horizontal asymptotes are lines that the graph of a function approaches as \( x \) heads towards positive or negative infinity.
- For rational functions, the horizontal asymptote can often be found by examining the degrees of the numerator and the denominator.
2. Evaluate the Function as \( x \to \infty \):
- Consider \( f(x) = \frac{1}{x + 3} \).
- To find the horizontal asymptote, we evaluate the limit of \( f(x) \) as \( x \) approaches infinity:
[tex]\[ \lim_{x \to \infty} \frac{1}{x + 3} \][/tex]
- As \( x \) becomes very large, the value of \( x + 3 \) also becomes very large. Thus, the term \(\frac{1}{x + 3}\) approaches 0.
3. Mathematical Result:
- So:
[tex]\[ \lim_{x \to \infty} \frac{1}{x + 3} = 0 \][/tex]
- This means that as \( x \) approaches infinity, the value of the function \( f(x) \) gets closer and closer to 0.
4. Conclusion:
- The horizontal asymptote of the function \( f(x) = \frac{1}{x + 3} \) is \( y = 0 \).
Thus, the correct answer is:
D. [tex]\( f(x) = 0 \)[/tex]
