Consider the function f(x) = (2x^3 - 5x^2 + x) / (3x^2 - 2x + 1). Find the limit of f(x) as x approaches positive infinity.
To find the limit of the function as x approaches positive infinity, we can analyze the degree of the numerator and denominator.
The highest power of x in the numerator is 3, while the highest power of x in the denominator is 2. Since the degree of the numerator is greater than the degree of the denominator, the limit at infinity may exist.
We will now divide both the numerator and denominator of the function f(x) by x^3, which is the highest power of x present:
f(x) = (2x^3 - 5x^2 + x) / (3x^2 - 2x + 1) = (2 - (5/x) + (1/x^2)) / (3/x - 2/x^2 + 1/x^3)
As x approaches positive infinity, the terms with negative powers of x approach zero. Thus, we can simplify the above expression:
lim(x→∞) f(x) = (2 - 0 + 0) / (0 - 0 + 0) = 2 / 0
The expression 2/0 is undefined. Therefore, the limit of f(x) as x approaches positive infinity does not exist.
Answer: The limit of f(x) as x approaches positive infinity does not exist.