Find limits using the division law

Learning Objectives State the formal Division Law for Limits. Apply the Division Law to find the limit of a rational function where the denominator's limit is non-zero. Identify when the Division Law for Limits cannot be directly applied (i.e., when the limit of the denominator is zero). Use algebraic factoring to simplify rational expressions that result in the indeterminate form 0/0. Evaluate the limit of a simplified rational function after canceling common factors. Distinguish between a limit that can be solved using the Division Law and one that requires algebraic manipulation first. How can we predict the final concentration of a chemical in a solution as one ingredient almost runs out? The answer lies in the powerful tool of limits for ratios! 🧪 This tutorial f...

TermDefinitionExample LimitThe value that a function f(x) approaches as the input x approaches some value 'a'. It describes the behavior of the function near a point, not necessarily at the point itself.The limit of f(x) = 2x as x approaches 3 is 6. Rational FunctionA function that can be written as the ratio of two polynomial functions, P(x) and Q(x), in the form f(x) = P(x) / Q(x), where Q(x) is not the zero polynomial.f(x) = (x^2 - 4) / (x - 2) is a rational function. Limit LawsA set of rules that allow us to calculate limits of complex functions by breaking them down into simpler parts (e.g., sums, products, quotients).The Sum Law states that the limit of a sum is the sum of the limits: lim [f(x) + g(x)] = lim f(x) + lim g(x). Division Law for LimitsA specific limit law stat...

The Division Law for Limits If \lim_{x \to a} f(x) = L and \lim_{x \to a} g(x) = M, where M ≠ 0, then: \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} = \frac{L}{M} Use this rule when you are finding the limit of a fraction (or rational function). The most important condition is that the limit of the denominator function must not be zero. Limit of a Polynomial Function If P(x) is a polynomial function, then: \lim_{x \to a} P(x) = P(a) This is the principle of direct substitution. To find the limit of a polynomial as x approaches 'a', you can simply plug 'a' into the function. This is used to find the limits of the numerator and denominator before applying the Division Law. Strategy for the Indeterminate Form 0/0 If...