To find the horizontal asymptote of the function f(x) = 3/(5x), we need to analyze the behavior of this function as x approaches infinity (both positively and negatively).
A horizontal asymptote represents a line that the graph of a function approaches but does not necessarily cross as x tends towards infinity or negative infinity.
1. **Analyzing f(x) as x approaches infinity:**
As x becomes very large (positive infinity), the term 5x dominates the denominator. Therefore, the function simplifies as follows:
f(x) = 3/(5x) → 0
This indicates that the horizontal asymptote is at y = 0, as the output of the function approaches 0 when x is large.
2. **Analyzing f(x) as x approaches negative infinity:**
Similarly, as x approaches negative infinity, the reasoning is the same. The denominator still approaches a significantly large negative number, leading us to:
f(x) = 3/(5x) → 0
Thus, the function also approaches 0 in this case as well.
Therefore, for both x → ∞ and x → -∞, the horizontal asymptote of the function f(x) = 3/(5x) is:
y = 0
In conclusion, the horizontal asymptote for the function is y = 0, which indicates that the graph will get arbitrarily close to the line y = 0 as x moves towards both positive and negative infinity.