Hello Engineer_Adam,

1.) lim(tan(x)/ln(cos(x)), x -> Pi/2)

If you plug in Pi/2 it's undefined;

L'Hopital's:

(tan(x))' = sec^2(x)

(ln(cos(x))' = -tan(x)

Take limit again; plug in Pi/2 --> undefined;

L'Hopital's:

(sec^2(x))' = (2*sin(x))/((cos(x))^3)

(-tan(x))' = -sec^2(x)

Repeat above;

Undef.

By now, it should be obvious that it will always be undefined regardless of how many times you take the derivative (the func. keeps getting messier and messier, in terms of cos in the denominator (and subsequently plugging in Pi/2 will give an undefined result).

2.) lim(x^(1/ln(x)), x -> 0)

Note: x^(1/ln(x)) = e; thus, the limit of the above is e.