Limits

[grim dove]

in the first hour find the distance he had climbed, same goes with others
like:
-1st hour: half the height of the mt, that means x = 1/2 x 1000
-....

[flat trout]

whole mountain is 1000m. to be quick lets say its 1 km
first hour it will go 1/2 km
second hour 1/4 km
third hour 1/8 km and it goes on.
so you need to find if it ever reaches 1 when you add them up
if you want to write it in limit form it should look like this (n=hour)

[zenith cave]

MathIsAlwaysRight

[silk kernel]

That's how far the climber can get with unlimited time. It's not immediately an answer to how long it will take.

(The problem is essentially a statement of Zeno's "race course" paradox, with some timing assumptions added).

The demand to "relate it to the concept of limits" does seem to require a bit of more or less dishonest handwaving of this kind, though.