Introduction to Limits

Pre-Calculus · Axiom Academy

A limit is the value a function heads toward as the input closes in on a point — even where the function never lands there. Picture walking toward a wall: each step halves the gap, yet you can talk about where you are headed long before you arrive. A limit captures exactly that — the value a function approaches as its input creeps toward a target, from either side. means f(x) gets arbitrarily close to L as x gets arbitrarily close to a — approaching from both the left and the right. 2. Notation, One Side at a Time We write the limit with a compact symbol. Read it left to right: "the limit of f(x) , as x approaches a , equals L " Sometimes we only let x creep in from one direction . A small superscript marks which side: — approach a from smaller values (the left). — approach a from larger values (the right). We can also ask what a function does as x grows without bound . This describes its end behavior — the height the curve settles toward far out to the right. means f(x) approaches L as x grows arbitrarily large. Here the curve flattens onto the line y=2 . A sequence is just a function whose inputs are the counting numbers . The same idea applies: the sequence runs , marching steadily toward 0 , so . The curve sits well above the line: at x=1 the height is 3 , still far from 2 . The term shrinks toward 0 , so the height squeezes down onto y=2 and stays there. You've seen a limit as the value a function approaches — from both sides, one side, or all the way out to infinity.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.