Properties of Limits: Apply It

  • Find the limit of a sum, a difference, and a product.
  • Find the limit of a polynomial.
  • Find the limit of a power or a root.
  • Find the limit of a quotient.

Analyzing Business Functions

Business models often use polynomial and rational functions to predict profit, cost, and revenue. Understanding limits helps us analyze what happens as production levels change, even at points where functions have discontinuities.

Question Help: Evaluating Limits Using Properties

  1. For polynomial functions, use direct substitution: [latex]\lim_{x \to a} p(x) = p(a)[/latex]
  2. For quotients that give [latex]\frac{0}{0}[/latex], try factoring and simplifying first
  3. For roots in quotients, consider multiplying by a conjugate
  4. Always try direct substitution first to see if it works

A company’s monthly profit in thousands of dollars is modeled by [latex]P(x) = 2x^3 - 3x + 1[/latex], where [latex]x[/latex] represents the number of units produced (in hundreds). As production approaches 500 units ([latex]x[/latex] approaches 5), what profit does the model predict?

Find [latex]\lim_{x \to 5} (2x^3 - 3x + 1)[/latex].

Polynomial functions are continuous everywhere, meaning the limit as [latex]x[/latex] approaches any value equals the function value at that point. This means we can always use direct substitution to evaluate their limits !

When direct substitution gives [latex]\frac{0}{0}[/latex], this is called an indeterminate form. It doesn’t mean the limit doesn’t exist—it means we need to simplify the expression algebraically first. The function [latex]\frac{x^2 - 6x + 8}{x - 2}[/latex] is equivalent to [latex]x - 4[/latex] everywhere except at [latex]x = 2[/latex] where it’s undefined.