Final answer:
The horizontal asymptotes of the function f(x) = √(2x²+1) / (2x-3) are y = 0.
Step-by-step explanation:
To find the horizontal asymptotes of the function f(x) = √(2x²+1) / (2x-3), we will examine the behavior of the function as x approaches positive infinity and negative infinity.
As x approaches positive infinity, the term with the higher power of x in the numerator and denominator will dominate. In this case, it is the square root term in the numerator and the x term in the denominator. Since the square root grows slower than x, the function will approach 0 as x approaches positive infinity. Therefore, the horizontal asymptote is y=0.
As x approaches negative infinity, the same reasoning applies. The square root term in the numerator and the x term in the denominator will dominate. Again, the square root grows slower than x, so the function will approach 0 as x approaches negative infinity. Therefore, the horizontal asymptote is also y=0.