Operators > Symbolic Operators > Example: Using the Limit Operator
Example: Using the Limit Operator
Using Infinity as a Limiting Value
1. Use the limit operator and symbolically evaluate an expression as its argument approaches infinity.
Click to copy this expression
2. Plot the function to facilitate its visualization. Use a horizontal marker to represent e.
Click to copy this expression
Click to copy this expression
Click to copy this expression
Click to copy this expression
Click to copy this expression
In the (x, y) quadrant we observe the following:
As n approaches positive infinity, the function approaches y=e.
As n approaches 0, the function approaches y=1.
Mathematically, this is represented by the following symbolic evaluations:
Click to copy this expression
Click to copy this expression
In the (-x, y) quadrant we observe the following:
As n approaches negative infinity, the function approaches y=e.
As n approaches -1, the function approaches y=infinity.
Mathematically, this is represented by the following symbolic evaluations:
Click to copy this expression
Click to copy this expression
* 
The use of Left-hand Limit-side in the second equation means that the -1 is to be approached from the left-side of the curve. If this is not specified, then the evaluation returns "undefined" because the function is not defined for -1 < n < 0:
Click to copy this expression
Using the Limit Side
1. Plot the cot function.
Click to copy this expression
Click to copy this expression
Click to copy this expression
Click to copy this expression
Click to copy this expression
In the (x, y) quadrant we observe the following:
As x approaches 0, the function approaches y=infinity.
As x approaches π, the function approaches y=-infinity.
Mathematically, this is represented by the following symbolic evaluations:
Click to copy this expression
Click to copy this expression
Since the function is symmetric around x=+/- n*π/2, the symbolic evaluation returns "undefined" because the function around x=0 (and any multiple number of π) can be either infinity or -infinity, depending on the side from which x approaches 0.
This is a good case for specifying the "Limit Side".
2. Specify the "Limit Side" and symbolically reevaluate the cot function around 0 and π.
Click to copy this expression
Click to copy this expression
Click to copy this expression
Click to copy this expression
The returned results agree with the plot.
* 
Sometimes it helps to plot a function in order to visualize it and to double check the validity of symbolic evaluation results.
Was this helpful?