I can't seem to figure out why the answers to these limits are what they are....can anyone help me understand where my thinking process is wrong:

lim (((e^-x)-2) / (e^1/x)) as x->0+

my thought process:

1. I try plugging in very small numbers that approach 0 into e^-x and see that the x in e^-x approaches 0, and that e^0 will equal 1.

2.for the denominator i try plugging some very small numbers in for x and see that e^(1/x) is e^infinity meaning e will get very big.

but the answer for this problem is 0. How?

I also struggle with this one:

lim x-> 0+ (e^-x) / (1 + e^(1/x)) = 0

I dont understand how the numerator will ever equal 0. e raised to any power will never make it equal zero....how is this the answer?