Find limits involving factorization and rationalization

Learning Objectives Identify when direct substitution for a limit results in the indeterminate form 0/0. Apply factorization techniques to simplify rational expressions within a limit problem. Calculate limits of rational functions by factoring and canceling common factors. Recognize when to use the conjugate to rationalize an expression involving a square root. Calculate limits of functions with radicals by rationalizing the numerator or denominator. Distinguish between limit problems requiring factorization and those requiring rationalization to resolve indeterminate forms. What happens when you try to plug a number into a function and get 0/0? Is it a dead end, or a hidden clue to the function's true behavior? 🤔 This tutorial explores the indeterminate form 0/0, wh...

TermDefinitionExample Indeterminate Form (0/0)A result obtained from direct substitution in a limit, such as 0/0, which does not provide enough information to determine the limit's value. It signals that further algebraic manipulation is required.For lim_{x->2} (x^2 - 4)/(x - 2), direct substitution gives (2^2 - 4)/(2 - 2) = 0/0. Direct SubstitutionThe initial method for evaluating a limit, where the value that x is approaching is plugged directly into the function. If this yields a real number, that is the limit.For lim_{x->3} (2x + 1), direct substitution gives 2(3) + 1 = 7. The limit is 7. FactorizationThe process of breaking down a polynomial into a product of its factors (simpler polynomials). This is used to find and cancel common factors in the numerator and denominator...

Difference of Squares a^2 - b^2 = (a - b)(a + b) This is the most common factorization pattern used in limit problems. It is also the underlying principle for why rationalization with a conjugate works. Conjugate Multiplication for Rationalization (\sqrt{a} - \sqrt{b})(\sqrt{a} + \sqrt{b}) = a - b To rationalize an expression containing a term like (√a - √b), multiply both the numerator and the denominator by its conjugate, (√a + √b), to eliminate the radicals. Limit of a Simplified Function If f(x) = g(x) for all x ≠ a, then lim_{x->a} f(x) = lim_{x->a} g(x) After factoring and canceling a term like (x-a), you create a new, simpler function g(x) that is identical to the original f(x) everywhere except at the point x=a. The limits of both functions as x approac...