Simple Limit

1) [cosx]=0 (exactly) |x|→0⁺ hence limit=1

3) put two cases , x>0 and x<0
for x>0 , 2.2^x ≥2√2 or, x≥1/2
for x<0 2^x + 2^-x >= 2√2 or , x ≤log₂ (√2-1)

in the second questn apply newtons leibentz theorem...
For differentiating x^x,
take log both sides...
ln(y) = x ln(x)
1ydydx=lnx+1
proceed further and i obtain d answer as ...

1-lny1+lny