Limit of a function - Indeterminate Form and LH rule

[Merz]

Homework Statement

Lim x-> 0 Sin(x) + Cos(x) + e^-x - 2 / Sin(x) - x

The Attempt at a Solution

Sin(x) + Cos(x) + e^-x - 2 = 0 and Sin(x) - x = 0

This is indeterminate form?

I used L'Hopitals rule 3 times to get

-Cos(x) + Sin(x) - e^-x / -Cos(x)

The top equals -2 and the bottom equals -1. The limit is 2. Am I understanding this right?

Homework Statement

Lim x->infinity (sqrt(x^2+1) - 1)^ln(x)

The Attempt at a Solution

I'm not sure what to do with ln(x). I believe this is the indeterminate form infinity - infinity.
Am I supposed to rewrite this someway?

 

[Anonymous217]

should be -sinx, not +sinx after doing it three times.
But yes, the answer is 2.