Yet another puzzle: now with graphs! [OVER]

Hi there! You like puzzles but bored with morse and binary translation, raging over vague and incomplete directions given or not having fun guessing the correct Egyptian or Greek god? Then welcome, maybe you'll find this one more entertaining.

Caution: some advanced math knowledge may be needed!

Ok, to the puzzle itself: assume we've got a function f(x) = 2 ^ (x / 4) - an exponential function with base 2 and power 0.25 * x. To solve the puzzle you need to answer the following questions:
1) Find out the distance between the origin point and the point where graph y = f(x) crosses the ordinate axis.
2) Using f(x) build a special function using this man's knowledge and find the limit of this function at minus infinity. You can also use his, his or his knowledge. All these men used such special functions but developed different notations for them.
3) Calculate the area under graph y = f(x) in II quadrant and get the function f(x) value using the area value obtained.
4) Find out abscissa of a point of crossing of graphs y = f(x) and y = x ^ (1 / x) in I quadrant.
5) Find out abscissa of a point of crossing of graphs y = f(x) and y = x. You need the integer one.

Now what to do with the answers: convert them to digits or letters (all the answers involved belong to the [-26; 26] interval; hope it's obvious how to convert them to letters). Oh, also there is a criterion on which letter is lowercase - if graph y = (x / 4) ^ 2 lies below y = f(x). Note: there is an intentional ambiguity how to interpret values belonging to the [-9; -1] U [1; 9] interval, if you get answers from this interval it's a little guessing work which are letters and which are numbers. Converting the answers to characters fill the X part of the link with them: http://www.steamgifts.com/giveaway/XXXXX.

Hope you will strug...eh... enjoy the puzzle! It will last for more than f(8) days. If I feel that the puzzle is too hard I'll give hints, but will do until there are, say, f(12) people in.

Solution:
1) Since this is a distance between origin and intersection with ordinate axis, you just need to find f(0) which is 1.
2) The man in the picture is Gottfried Leibnitz, who (alongside Newton) greatly developed infinitesimal calculus. His notation for function differentiation is most commonly used. Other people on the pictures (Lagrange, Newton and Euler) developed own notations for differentiation. So the task was to find a derivative of f(x) and then its limit at minus infinity. Observant people should notice that since the function is exponential with positive coefficient before independent variable all its derivatives as well as anti-derivatives also should be similar exponential functions with positive coefficient before independent variable. Hence finding derivative wasn't necessary as all derivatives as well as f(x) itself have a limit 0 at minus infinity.
3) To find area under f(x) in II quadrant one should find a value of definite integral of f(x) from minus infinity to zero. This value turns out to be 4 / ln(2), where ln is a natural logarithm. Since 4 / ln(2) = 4 * log2(e), where log2 is a logarithm with base 2, value of f(4 / ln(2)) = 2 ^ ((4 / ln(2)) / 4) = 2 ^ (1 / ln(2)) = 2 ^ (log2(e)) = e. It's lowercase because f(e) > 1 (since e / 4 > 0) and (e / 4) ^ 2 < 1 (since e / 4 < 1).
4) Graphs y = f(x) and y = x ^ (1 / x) cross twice in I quadrant, at x = sqrt(2) and x = 2. Both roots should hint answer 2.
5) Graphs y = f(x) and y = x again cross twice, once at a point with transcendental abscissa and at x = 16. So the answer to the question is P (16th letter). It is uppercase because f(16) = 16 and (16 / 4) ^ 2 = 16 and therefore the criterion doesn't hold true at this point.

The resulting giveaway link is http://www.steamgifts.com/giveaway/10e2P. Thanks for all who solved and tried to solve the puzzle. Congratulations to the winner and good luck to everyone else!