For the function f(x) = 2x^3 - 5x^2 + 3x - 1, as x approaches negative infinity, f(x) approaches _[blank]_.
Can you explain step by step please?
To find out what the function approaches as x approaches negative infinity, we need to evaluate the limit of the function as x approaches negative infinity.
Step 1: Substitute negative infinity (-∞) for x in the function f(x).
f(-∞) = 2(-∞)^3 - 5(-∞)^2 + 3(-∞) - 1.
Step 2: Simplify any terms that involve negative infinity.
Since we have negative infinity raised to a power, we can simplify the expression as follows:
For any positive number n, -∞ to the power of n is negative infinity (-∞), and -∞ to the power of an odd negative number is negative infinity (-∞).
Therefore, we have:
f(-∞) = 2(-∞) - 5(-∞)^2 + 3(-∞) - 1.
Step 3: Simplify the terms.
Since we are dealing with negative infinity, we consider the dominant term in the function, which is the term with the highest exponent.
In this case, the term with the highest exponent is -5x^2. When x approaches negative infinity, x^2 becomes very large, and the term -5x^2 dominates the expression.
Thus, we can ignore the remaining terms and simplify the expression to -5(-∞)^2 = -5∞^2.
Step 4: Determine the value of the expression.
Since -5∞^2 does not have a defined value, it is undefined.
Therefore, as x approaches negative infinity, f(x) approaches undefined.