Explore graphically the limit

for various values of a and b.

In the following LiveMath Notebook, you can see the graph of

for a = 2 and b = 4 in blue. If L is the limit above, the red line is the graph of y = L + e and the green line is the graph of y = L - e.

By making the values of e smaller positive numbers, see if you can determine graphically what is the value of L.

Explore with changing a and b; consider (a, b) = (1, 1), (1, 2), (-2, 1), and (6, 2) and calculating the corresponding limits L. Can you detect a pattern so that you can evaluate L without knowing the particular values of a and b?