To determine the vertical and horizontal asymptotes of the function f(x) = 3x² + 2x + 4, we need to analyze the function’s limits and behavior as x approaches certain values.
Vertical Asymptotes
Vertical asymptotes occur when the function approaches infinity or negative infinity as x approaches a specific value. These typically occur at values of x where the function is undefined, usually due to division by zero in rational functions. However, since f(x) = 3x² + 2x + 4 is a polynomial function, it is defined for all real numbers.
Thus, f(x) does not have any vertical asymptotes.
Horizontal Asymptotes
Horizontal asymptotes describe the behavior of a function as x approaches positive or negative infinity. To find horizontal asymptotes, we look at the limits:
1. As x → ∞:
In this case, the dominant term in the polynomial is 3x². As x increases indefinitely, the other terms (2x and 4) become insignificant in comparison. Thus, we have:
lim (x → ∞) f(x) = 3x² → ∞
2. As x → -∞:
Similarly, for negative infinities, the dominant term will once again be 3x², leading us to the same conclusion:
lim (x → -∞) f(x) = 3x² → ∞
Since the function approaches infinity in both directions, f(x) does not have a horizontal asymptote either.
Conclusion
In summary, for the function f(x) = 3x² + 2x + 4:
- No vertical asymptotes
- No horizontal asymptotes