AP Calculus AB Exam Question:

Let f(x) = 2x^3 - 5x^2 + 3x + 1.

Determine the following limit at infinity:

lim(x->∞) f(x)

Step-by-step Solution:

To find the limit of f(x) as x approaches infinity, we evaluate the behavior of the function as x gets larger and larger.

First, let's examine the dominant term in the function, which is 2x^3. As x approaches infinity, the term 2x^3 dominates the function.

Now, divide every term in the function f(x) by x^3:

f(x) = 2x^3/x^3 - 5x^2/x^3 + 3x/x^3 + 1/x^3

Simplifying this expression, we get:

f(x) = 2 - 5/x + 3/x^2 + 1/x^3

As x approaches infinity, the terms with higher powers of x tend to zero, leaving only the constant term 2. Therefore, the limit of f(x) as x approaches infinity is simply:

lim(x->∞) f(x) = 2

Answer: The limit of f(x) as x approaches infinity is 2.

Note: This solution assumes knowledge of polynomial functions and their behavior as the degree of the polynomial increases.