To compute a limit, you must specify a functional expression, limit a variable, and a limiting value for that variable. If you do not specify a direction, Axiom attempts to compute a two-sided limit.
Issue this to compute the limit
Sometimes the limit when approached from the left is different from the limit from the right and, in this case, you may wish to ask for a one-sided limit. Also, if you have a function that is only defined on one side of a particular value, limit:one-sided vs. two-sided you can compute a one-sided limit.
The function is only defined to the right of zero, that is, for . Thus, when computing limits of functions involving , you probably want a ``right-hand'' limit.
When you do not specify `` '' or `` '' as the optional fourth argument, limit tries to compute a two-sided limit. Here the limit from the left does not exist, as Axiom indicates when you try to take a two-sided limit.
A function can be defined on both sides of a particular value, but tend to different limits as its variable approaches that value from the left and from the right. We can construct an example of this as follows: Since is simply the absolute value of , the function is simply the sign ( or ) of the nonzero real number . Therefore, for and for .
This is what happens when we take the limit at . The answer returned by Axiom gives both a ``left-hand'' and a ``right-hand'' limit.
Here is another example, this time using a more complicated function.
You can compute limits at infinity by passing either limit:at infinity or as the third argument of limit.
To do this, use the constants and .
You can take limits of functions with parameters. limit:of function with parameters As you can see, the limit is expressed in terms of the parameters.
When you use limit, you are taking the limit of a real function of a real variable.
When you compute this, Axiom returns because, as a function of a real variable, is always between and , so tends to as tends to .
However, as a function of a complex variable, is badly limit:real vs. complex behaved near (one says that has an essential singularity essential singularity at ). singularity:essential
When viewed as a function of a complex variable, does not approach any limit as tends to in the complex plane. Axiom indicates this when we call complexLimit.
Here is another example. As approaches along the real axis, tends to .
However, if is allowed to approach along any path in the complex plane, the limiting value of depends on the path taken because the function has an essential singularity at . This is reflected in the error message returned by the function.
You can also take complex limits at infinity, that is, limits of a function of as approaches infinity on the Riemann sphere. Use the symbol to denote ``complex infinity.''
As above, to compute complex limits rather than real limits, use complexLimit.
In many cases, a limit of a real function of a real variable exists when the corresponding complex limit does not. This limit exists.
But this limit does not.