Formal Limit as x appoaches +- infinity
- Author:
- Dr. Jack L. Jackson II
End Behavior
"End Behavior" is a misnomer, because many graphs do not have an end; they go on forever.
By end behavior we mean to examine what happens on the graph as the x-values get larger and larger without bound (x approaches infinity) and also when the x-values get less and less without bound (x-approaches negative infinity). If one of these limits (L) exists, then the graph approaches a horizontal line (y = L) on that side. Here is the formal definition of the limit as x approaches infinity.
Definitions:
Given a function f and a real number L, we say that if and only if
for any > 0 there exist M > 0 such that
if x > M then L - < < L + .
Given a function f and a real number L, we say that if and only if
for any > 0 there exist M > 0 such that
if x < -M then L - < < L + .
In the App:
Enter the formula for a desired function in the input box for f(x). It will be graphed.
Enter a value for the limit on the right side by typing in the input box for L.
A horizontal line at y = L will be graphed. If this is the correct value for L, then the graph of f(x) will get closer and closer to this line on the right sided of the graph. You may have to zoom out to see this.
Move the point M so that for every x-value to the right of x = M the graph of f(x) is in the shaded box with y-values within epsilon of L.
For the left limit, repeat the above with a potentially different value in L. This time we want the graph to be in the box on the left to the left of -M. We move the point at M to control -M.