Find limits of polynomials and rational functions

Learning Objectives Evaluate the limit of any polynomial function using direct substitution. Evaluate the limit of a rational function at a point where the denominator is non-zero. Identify the indeterminate form 0/0 when evaluating limits of rational functions. Apply algebraic techniques, such as factoring and cancelling, to resolve the indeterminate form 0/0. Determine the limit of a rational function as x approaches positive or negative infinity by analyzing the degrees of the numerator and denominator. Distinguish between limits that approach a finite number, infinity, negative infinity, or do not exist. What happens to the value of a function as you get infinitely close to a specific point, without ever actually touching it? 🤔 This tutorial will equip you with the fun...

TermDefinitionExample LimitThe value that a function f(x) approaches as the input x approaches some value c. We write this as lim_{x->c} f(x) = L.For the function f(x) = x + 2, as x gets closer to 3, f(x) gets closer to 5. So, lim_{x->3} (x + 2) = 5. Polynomial FunctionA function consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.P(x) = 4x^3 - 2x^2 + x - 10 Rational FunctionA function that can be expressed as the ratio of two polynomial functions, R(x) = P(x) / Q(x), where the denominator Q(x) is not the zero polynomial.R(x) = (x^2 - 4) / (x + 1) Direct SubstitutionThe primary method for finding limits of well-behaved functions (like polynomials) by simply plugging the valu...

Limit of a Polynomial Function If P(x) is a polynomial function and c is any real number, then: lim_{x->c} P(x) = P(c) To find the limit of any polynomial at any point, always use direct substitution. Polynomials are continuous everywhere, so the limit is always the function's value at that point. Limit of a Rational Function If R(x) = P(x)/Q(x) is a rational function and c is a real number such that Q(c) ≠ 0, then: lim_{x->c} R(x) = P(c) / Q(c) Use direct substitution for rational functions as long as the denominator does not become zero. If the denominator is zero, you must use other techniques. Limits of Rational Functions at Infinity For R(x) = (a_n x^n + ...)/(b_m x^m), the limit as x->±∞ is determined by the degrees n and m: 1. If n < m, the lim...