Real-Valued Limits
A mathematical limit is a concept that informs mathematicians of values toward which a function f(x) tends as x approaches some value c. Limits are useful because when they exist, they allow one to determine, without directly evaluating f(c), the behavior of f(x) near c, regardless of whether the function is defined at that point.
Approaching 3 from the left and the right yields values extremely close to f(3), so the limit exists at x = 3.
Here is the graph of the function f(x) = 1/x. If we want to determine its behavior at x = 0, we must find the limit as x goes to zero from both directions.
Approaching x = 0 from the left causes f(x) to tend to negative infinity.
Approaching x = 0 from the right causes f(x) to tend to positive infinity.
Since the directions do not agree, the limit of f(x) does not exist at x = 0.
Approaching x = 0 from the left causes f(x) to tend to negative infinity.
Approaching x = 0 from the right causes f(x) to tend to positive infinity.
Since the directions do not agree, the limit of f(x) does not exist at x = 0.
Complex-Valued Limits
In the real number system, we can approach some value c from two directions, left and right, because our inputs are confined to a single axis.
The complex numbers, however, exist on a plane. Therefore, there are infinitely many directions from which we can approach some input z0.
The complex numbers, however, exist on a plane. Therefore, there are infinitely many directions from which we can approach some input z0.
We can understand how to approach thinking about complex limits mathematically by simplifying the process. Instead of approaching from infinitely many directions, let approach z0 from the real axis and keep our imaginary direction constant. This will give us some value for w0.
Now, let us approach z0 from the imaginary axis and keep our real direction constant. This will also give us some value for w0.
If the two limit values obtained do not agree, then the limit does not exist. If the two directions do agree, then the limit does exist.