Infinite Limits and Limits at Infinity

Here we plot the function f(x) = which has a discontinuity at x = 2. Specifically, the function is undefined at this value. We first define the function by

> f := x -> 3*x/(x-2);

Now let's try to plot this with the plot command. Try the following three options sequentially to get a better picture of this graph.

> plot(f(x),x=-4..8,labels=["x","f"]);

This shows some peculiar behavior near x = 2. We reduce the range (y values) by restricting the y-values with the command y=-15..15.

> plot(f(x),x=-4..8,y=-15..15,labels=["x","f"]);

We now get a better picture from this but there appears to be a vertical line at x = 2. We know this cannot be the case because the function is undefined there. We let Maple know there is a discontinuity with the following command. You need not specify where the discontinuity lies.

> plot(f(x),x=-4..8,y=-15..15,discont = true,labels=["x","f"]);

The above sequence of graphs should make it clear that the limit of f(x) as x approaches 2 from the left is

negative infinity (- ) and the limit of f(x) as x approaches 2 from the right is positive infinity ( ). These are called infinite limits and describe an infinite limit of the function as x approaches a finite value (2).

Does exist? Try f (1.99999) and f (2.00001).

Maple has a command for finding these limits.

> limit(f(x),x=2);

This "undefined" makes sense because the limit from the left is different from the limit from the right. We can determine these limits from the left and right by including this in the limit command. The command below finds the limit from the left.

> limit(f(x),x=2,left);

Similarly the limit from the right of x = 2 is found by

> limit(f(x),x=2,right);

Therefore we say: "the limit of f as x approaches 2 from the left is negative infinity and the limit of f as x approaches 2 from the right is (positive) infinity. Because these two are not the same, "the limit of f as x approaches 2 is undefined".

It also appears that as x goes to negative and positive infinity, the function tends towards a finite value. I initially guess this value to be 3. I check this by increasing my domain and plotting the line y=3 as well.

> plot([f(x),3],x=-15..15,y=-15..15,discont = true,labels=["x","f"]);

It is now clear that as x goes to positive or negative infinity, the function tends to 3. These are called limits at infinity. You state: "The limit as x approaches positive or negative infinity is 2".

Problem number 1 asks for these same types of limits for a different function.