Observe in the graph at the right that the graph of the original function and the graph of the function that results from applying L'Hopital's Rule are not the same but both have the same limit as x approaches 2.  The original function is undefined at x = 2.  The function produced by applying L'Hopital's Rule, g, is not and g(2) = 5/3.

The limit does appear to be 2 based on the graph at the right.

The limit does not appear to exist based on the graph at the right.

The limit does appear to be 0 based on the graph at the right.

The limit does appear to be 0 based on the graph at the right.

The limit does appear to be e based on the graph at the right.

The horizontal blue line is the graph of y = e.

The limit does appear to be 1 based on the graph at the right and the numerical analysis above.  Maple is in agreement.

There is no way for this integral to converge when, after the u-substitution, the argument for the new integral (u) is not approaching zero as u approaches infinity.

Since the limit above is not zero, the integral must diverge.

The improper integral "diverges" at both ends.

Integration of Sec³x

Extra Credit:  Find the volume (of the horn).