This is always a challenging topic for Calculus students. My approach is to introduce the definition, draw a picture of the definition and then have the students work through this worksheet.
The worksheet starts by asking students to find values of delta for given values of epsilon using a linear function. This exploration helps students understand what is happening before they are asked to find the value of delta in terms of epsilon on the next part of the worksheet.
Next the students explore a question about tolerance of error, which opens the discussion on how error relates to epsilon and delta.
Lastly, the definition of a limit approaching infinity is introduced followed by questions on this definition given a rational function.