sinh is the hyperbolic sine, and I'm surprised it's in your calc 1 course

from wikipedia:

sinh x = {e^x - e^{-x}} / {2} = {e^{2x} - 1} / {2e^x}

so sinh x / e^x = (e^2x - 1) / (2e^2x)

and lim as x-> 0 is equivalent to (e^2(0) - 2) / (2e^2(0))

anything to the zeroth power is 1, so this is (1 - 2) / (2), or -.5

Regarding four:

4. Let f(x)=

-x+2 for x≤1

x^2-2x+2 for x>1

The definition of differentiability at a point is that it's a smooth curve at that point

This wouldn't create a smooth curve, which you can see by drawing the two functions; you'll notice a sharp change from a downwards line to a upward curve

I don't know how to show that mathematically without taking derivatives, unfortunately