Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the first step to evaluate a limit with a higher exponent in the denominator?

1. Multiply the numerator by (1/xⁿ)

2. Divide the top and bottom by (1/xⁿ)

3. Simplify the expression directly

4. Use L'Hôpital's Rule immediately

The correct answer is: Divide the top and bottom by (1/xⁿ)

To evaluate a limit where there is a higher exponent in the denominator, dividing both the numerator and denominator by \(1/x^n\) can simplify the expression significantly. This technique allows for easier analysis of how the numerator and denominator behave as the variable approaches the limit value, especially when the variable approaches infinity. By applying this method, the terms of the numerator that are of lower degree than the denominator can effectively be highlighted or eliminated. This simplification helps in determining the limit more straightforwardly without directly resorting to more complex methods like L'Hôpital's Rule or unnecessary algebraic manipulation. This approach is particularly useful when dealing with polynomial functions, as it can reveal the dominant term in both the numerator and denominator, guiding towards the correct limit. Using this method sets the stage for a clearer evaluation of the expression and allows for better insight into the behavior of functions at critical points.