Represent numbers (up to 20)

Learning Objectives Define a limit as a value that a function or sequence approaches. Represent an integer (up to 20) as the limit of a sequence as n approaches infinity. Represent an integer (up to 20) as the limit of a function at a specific point. Represent an integer (up to 20) as the sum of a convergent infinite geometric series. Evaluate one-sided limits that converge to an integer value. Differentiate between a function's value at a point, f(c), and its limit, L, using integer examples. Apply formal limit notation to describe how a value is being approached. How can we express the number 10 without ever actually writing the number 10 itself? 🤔 Let's use calculus to get infinitely close! This tutorial introduces the fundamental concept of limits. We will...

TermDefinitionExample LimitThe value that a function or sequence 'approaches' as the input or index approaches some value (which could be infinity). It describes the intended destination, not necessarily the actual value at that point.The limit of the function f(x) = 2x as x approaches 4 is 8. We write this as lim_{x→4} 2x = 8. SequenceAn ordered list of numbers, often following a specific pattern or rule, denoted by a_n.The sequence a_n = 5 - 1/n generates the terms 4, 4.5, 4.66..., 4.9, 4.99, ... which approaches the number 5. ConvergenceA property of an infinite sequence or series whose terms approach a specific finite number, known as the limit.The sequence a_n = 1/n converges to 0 as n approaches infinity. Infinite Geometric SeriesThe sum of the terms of an infinite geometr...

Limit of a Sequence (approaching an integer) lim_{n→∞} (k - c/n) = k Used to show a sequence approaching an integer 'k'. As 'n' becomes infinitely large, the fraction 'c/n' becomes infinitesimally small (approaches 0), leaving 'k' as the limit. Limit of a Polynomial Function lim_{x→c} P(x) = P(c) For any polynomial function P(x), the limit as x approaches a number 'c' can be found by direct substitution of 'c' into the function. Sum of a Convergent Infinite Geometric Series S = a / (1 - r), where |r| < 1 Used to find the finite sum 'S' of an infinite geometric series, where 'a' is the first term and 'r' is the common ratio. This is a powerful way to represent an integer as a...