A Formal Definition Topics: Explorations>Lectures>Practice

Section 1.2 - Limits, a Formal Definition

Our informal definition was based on some very loose language like "close to". In this lecture, we're going to make that more precise. From a high level, the idea is that if there's a point, y, close to the limit on the y-axis then there has to be a corresponding point close to the limit value on the x-axis that the function maps to that point. This is illustrated in the exploration below.

The function on the graph is $f(x) = 1.25x - .5$ and we're looking at $\lim\limits_{x\to 1} f(x)$. I'm going to claim that the limit is equal to $f(1) = .75$. The directions below will walk you through our new definition of "close to".

ε	y Interval	x Interval	δ

Pick a value for "ε". Start with something small, say in the .5 to .75 range. That's going to define an interval around our proposed limit, .75.

ε =
Now you can see the range of y values bracketing .75: .75 + ε above it and .75 - ε below it.
Next calculate the x-values that correspond to the y-values at the ends of the interval.
Notice how the new interval on the x-axis made by those points brackets the x value in our limit - the same distance on either side. What is that distance?
What we can see here is that an interval of ±δ "corresponds"