Which function has no horizontal asymptote?
A f(x) = 2x-1 3×2 x-1
B f(x) = 3x 2×2
C f(x)=3x-1
The Correct Answer and Explanation is :
To determine which function has no horizontal asymptote, we need to analyze each function’s behavior as ( x ) approaches infinity (or negative infinity). Horizontal asymptotes indicate the value that a function approaches as the input becomes very large or very small. Let’s examine each function provided:
Function A: ( f(x) = \frac{2x – 1}{3x^2 + x – 1} )
To find the horizontal asymptote, we consider the leading terms of the numerator and the denominator:
- The leading term of the numerator is ( 2x ).
- The leading term of the denominator is ( 3x^2 ).
As ( x ) approaches infinity, the function behaves like:
[
f(x) \approx \frac{2x}{3x^2} = \frac{2}{3x} \to 0 \quad \text{(as } x \to \infty\text{)}
]
Thus, this function has a horizontal asymptote at ( y = 0 ).
Function B: ( f(x) = \frac{3x}{2x^2} )
Similarly, we analyze this function:
- The leading term of the numerator is ( 3x ).
- The leading term of the denominator is ( 2x^2 ).
As ( x ) approaches infinity, the function behaves like:
[
f(x) \approx \frac{3x}{2x^2} = \frac{3}{2x} \to 0 \quad \text{(as } x \to \infty\text{)}
]
This function also has a horizontal asymptote at ( y = 0 ).
Function C: ( f(x) = 3x – 1 )
This function is a linear function. As ( x ) approaches infinity, the function behaves like:
[
f(x) \to 3x – 1 \to \infty \quad \text{(as } x \to \infty\text{)}
]
Since the output grows indefinitely without approaching a specific finite value, this function does not have a horizontal asymptote.
Conclusion
The function that has no horizontal asymptote is C: ( f(x) = 3x – 1 ). Linear functions such as this one typically do not have horizontal asymptotes because their value increases or decreases indefinitely as ( x ) approaches infinity or negative infinity, in contrast to rational functions which often settle at specific values determined by their leading coefficients.