Final answer:
The limit as x approaches infinity of f(x) is 3, and the limit as x approaches negative infinity of f(x) is also 3. The values of x₁ and x₂ cannot be determined.
Step-by-step explanation:
To find the limit as x approaches infinity, we can simplify the function by dividing both the numerator and denominator by the highest power of x. In this case, the highest power is x, so we divide the numerator and denominator by x.
Dividing 3x by x gives us 3, and dividing √(x²+4) by x gives us √((x²+4)/x²), which simplifies to √(1+4/x²). As x approaches infinity, 4/x² approaches 0, and √(1+0) is equal to 1.
Therefore, the limit as x approaches infinity of f(x) is 3/1 = 3.
To find the limit as x approaches negative infinity, we can use the same process. Dividing 3x by x gives us 3, and dividing √(x²+4) by x gives us √((x²+4)/x²), which simplifies to √(1+4/x²), which again approaches 1 as x approaches negative infinity.
Therefore, the limit as x approaches negative infinity of f(x) is 3/1 = 3.
The values of x₁ and x₂ in terms of f(x) are not provided in the question, so they cannot be determined.