Okay, so I need to find the horizontal asymptote of the function y = (4x³ + 2x - 4)/(3x³ - 2x² + 5x - 1). Hmm, horizontal asymptotes... I remember they have something to do with the behavior of the function as x approaches infinity or negative infinity. Let me think. 

First, horizontal asymptotes are determined by the degrees of the numerator and the denominator. If the degrees are the same, then the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is less than the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is higher, there's no horizontal asymptote, but maybe an oblique asymptote instead. Wait, in this case, both numerator and denominator are degree 3. So, the horizontal asymptote should be the ratio of the leading coefficients. 

Let me verify that. The numerator is 4x³ + 2x - 4. The leading term is 4x³. The denominator is 3x³ - 2x² + 5x - 1. The leading term is 3x³. So, the ratio of the leading coefficients is 4/3. Therefore, the horizontal asymptote should be y = 4/3. 

But wait, let me make sure I'm not missing anything here. Sometimes, when there are higher degree terms, things can cancel out or something. But in this case, both are degree 3, so when x becomes very large, the lower degree terms become negligible. So, the behavior of the function is dominated by the 4x³ and 3x³ terms. Therefore, dividing both numerator and denominator by x³, we get:

y = (4 + 2/x² - 4/x³) / (3 - 2/x + 5/x² - 1/x³)

As x approaches infinity, all those terms with 1/x, 1/x², etc., go to zero. So, we're left with y = 4/3. That seems right. 

But just to double-check, maybe I can plug in a really large x value and see if the function approaches 4/3. Let's take x = 1000. Then the numerator is approximately 4*(1000)^3 = 4,000,000,000. The denominator is approximately 3*(1000)^3 = 3,000,000,000. So, 4/3 is exactly 1.333..., and the actual value would be 4,000,000,000 / 3,000,000,000 = 1.333..., which is 4/3. So, even with a finite large x, it's already exactly 4/3. That makes sense because the lower terms don't affect the limit as x approaches infinity. 

Alternatively, if I factor out x³ from numerator and denominator:

Numerator: x³(4 + 2/x - 4/x²)
Denominator: x³(3 - 2/x + 5/x² - 1/x³)
Then, y = [x³(4 + ...)] / [x³(3 + ...)] = (4 + ...)/(3 + ...) as x approaches infinity. The ... terms go to zero, so y approaches 4/3. 

Therefore, I think the horizontal asymptote is definitely y = 4/3. 

Wait, but let me check if there's any possibility that the function could approach a different value. For example, sometimes when there are higher degree terms, but in this case, both numerator and denominator are degree 3, so leading coefficients are the key. If the degrees were different, the horizontal asymptote would be different. But here, same degree, so ratio of leading coefficients. 

I don't see any mistakes in my reasoning. So, the answer should be y = 4/3. 

**Final Answer**
The horizontal asymptote is at \boxed{\dfrac{4}{3}}.
Therefore, there is a horizontal asymptote at $y = \boxed{\frac{4}{3}}$.
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