- The function is \(f(x) = \frac{3x}{4-x}\).
- Divide both the numerator and the denominator by \(x\) to simplify the expression: \(\frac{3}{4/x - 1}\).
- As \(x \to \infty\), \(\frac{4}{x} \to 0\), so the limit is \(\frac{3}{0 - 1} = -3\).
- Final Answer: The graph approaches \(\boxed{-3}\) as \(x\) approaches infinity.
Explanation
- Problem Analysis
The function given is \(f(x) = \frac{3x}{4-x}\). This is a rational function, which is the ratio of two polynomials: the numerator \(p(x) = 3x\) and the denominator \(q(x) = 4-x\). We want to determine the behavior of this function as \(x\) approaches infinity, which corresponds to finding the horizontal asymptote.
- Evaluating the Limit
To find the limit \(\lim_{x \to \infty} \frac{3x}{4-x}\), we divide both the numerator and the denominator by the highest power of \(x\) present in the expression, which is \(x^1\).
Applying this division:
\[\lim_{x \to \infty} \frac{3x}{4-x} = \lim_{x \to \infty} \frac{\frac{3x}{x}}{\frac{4-x}{x}} = \lim_{x \to \infty} \frac{3}{\frac{4}{x} - 1}\]
As \(x \to \infty\), the term \(\frac{4}{x}\) approaches 0. Substituting this value, the limit becomes:
\[\frac{3}{0 - 1} = \frac{3}{-1} = -3\]
- Conclusion and Final Answer Selection
Based on our calculation, the function \(f(x) = \frac{3x}{4-x}\) approaches \(-3\) as \(x\) approaches positive infinity. Comparing this result with the given choices, the statement "The graph approaches -3 as x approaches infinity" is correct.
Examples
In engineering, analyzing the long-term behavior of a system is crucial. For instance, consider a chemical reactor where the concentration of a product, \(C(t)\), changes over time \(t\) according to a model \(C(t) = \frac{3t}{4+t}\) (or similar rational functions). As time approaches infinity, the concentration levels off, representing the steady-state value of the system. Understanding these limits helps engineers predict when the system reaches stability and what that final state will be.