Can someone please explain what I am studying when I am studying limits, delta & epsilon. I am having a really difficult time visualizing what is going on and what these things represent… Not so much with limits. I “know” what a limit is„, but how they all tie in together & the other two specifically. If someone could help I would be ETERNALLY grateful!

Tuesday, September 6, 2011

proto-flake asked: Can someone please explain what I am studying when I am studying limits, delta & epsilon. I am having a really difficult time visualizing what is going on and what these things represent... Not so much with limits. I "know" what a limit is,,, but how they all tie in together & the other two specifically. If someone could help I would be ETERNALLY grateful!

Often, the first visual interpretation of a limit is one in which you picture moving along a function by changing your x value, and for a function that seems to have no holes, the simplest way to find the value of a limit is to substitute the target value for x and see what the output value of the function is. However, this is not a rigorous interpretation because, often, there is an obstacle (like an undefined solution) with the output value. The ε-δ definition allows us to define regions about the point you’re evaluating, and it ultimately allows us to have rigorous definitions for when a limit does and does not exist.

The δ in the definition applies to a region about the x value in question (say, a) — the region is defined by a - δ ≤ a ≤ a + δ. The ε in the definition applies to a region about the function’s output value, say, L: L - ε ≤ L ≤ L +ε.

The definition itself corresponds to what may be interpreted as a sort of game. If you have two players — the “devil’s advocate” on one side, on the y-axis, and you on the x-axis — then it’s the devil’s advocate who claims that you can’t bring the true value of the limit to be in the region of L - ε ≤ L ≤ L + ε. In response, you adjust your value of δ to allow the limit to exist in that region, and on his turn, he shrinks the region even further, by decreasing the value of ε. If the limit is to exist, then as long as ε and δ remain greater than 0, you will always be able to achieve a limit value within the given ranges (ε and δ need to be kept greater than 0 because the goal is to define the limit based on the immediate region about the function, not evaluate at a itself — such would be simply substituting a into f(x)). Hence come the familiar relations that govern the ε-δ definition: Given that