Delta Epsilon Limit Definition

The Epsilon-Delta Definition of Limits

The Epsilon-Delta Definition of Limits
The epsilon-delta definition of limits allows a method of finding values of the function that are close enough to the limit to indicate where the function is heading, even in undefined sections of the graph. First let’s look at how the definition works graphically:

Our function, f(x), is shown by the blue line. L is the limit as x approaches c. We are using two Greek letters, ε (epsilon) and δ (delta). The error tolerance, ε, establishes the distance from the limit by restricting the range of the function from L+ε to L-ε. The domain of the function from c is restricted by the variable δ. Values of x must be between, but not actually at, c+δ and c-δ. By determining c+δ and c-δ, the boundaries for the domain are established. We create the yellow range of the graph by shading the graph δ units from either side of c. Substituting values within this domain for x will result in y-values within the green range. By substituting c+δ and c-δ into the function, the results will determine the range of the limit, the blue section of the graph. The bounds of the range of the limit are denoted by L+ε and L-ε. Once the blue and yellow boundaries are established, it is then possible to determine the limit by observing the x and y values of the function within the green section (where blue and yellow intersect). Any values for x that are within the yellow range can be substituted into the function to find y-values in the green area. The y-values in this green area indicate the limit.

Now let us consider the epsilon-delta definition from an algebraic perspective.

lim
→

=L

For all ε > 0, there exists a δ > 0 such that: If