To identify the asymptotes and state the end behavior for the function
\[ f(x) = \frac{3x}{x - 9} \]
we will look for vertical asymptotes, horizontal asymptotes, and determine the end behavior of the function.
Step 1: Identify Vertical Asymptotes
Vertical asymptotes occur where the denominator is equal to zero, provided the numerator is not also zero at that point.
Set the denominator equal to zero and solve for \(x\):
\[ x - 9 = 0 \implies x = 9 \]
Thus, there is a vertical asymptote at \(x = 9\).
Step 2: Identify Horizontal Asymptotes
To find the horizontal asymptote, we analyze the end behavior of the function as \(x\) approaches infinity (\(x \to \infty\) or \(x \to -\infty\)). We do this by considering the degrees of the numerator and denominator:
- The degree of the numerator \(3x\) is 1.
- The degree of the denominator \(x - 9\) is 1.
Since the degrees of the numerator and denominator are the same, we can find the horizontal asymptote by taking the ratio of the leading coefficients.
The leading coefficient of the numerator (3) divided by the leading coefficient of the denominator (1) gives us:
\[ y = \frac{3}{1} = 3 \]
Thus, there is a horizontal asymptote at \(y = 3\).
Step 3: Determine End Behavior
Next, we examine the end behavior of the function:
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As \(x \to \infty\):
\[ f(x) = \frac{3x}{x - 9} \] As \(x\) becomes very large, \(-9\) becomes negligible compared to \(x\), so:
\[ f(x) \approx \frac{3x}{x} = 3 \] Therefore, as \(x \to \infty\), \(f(x) \to 3\).
-
As \(x \to -\infty\):
We analyze:
\[ f(x) = \frac{3x}{x - 9} \] Similar to the previous case, as \(x\) becomes very negative, \(-9\) becomes negligible compared to \(x\), so:
\[ f(x) \approx \frac{3x}{x} = 3 \] Thus, as \(x \to -\infty\), \(f(x) \to 3\).
Summary
- Vertical Asymptote: \(x = 9\)
- Horizontal Asymptote: \(y = 3\)
- End Behavior: As \(x \to \infty\), \(f(x) \to 3\); as \(x \to -\infty\), \(f(x) \to 3\).
In conclusion, the function approaches the horizontal asymptote \(y = 3\) from both sides of the vertical asymptote at \(x = 9\).