two extremely difficult limit problems

[fankafoon]

1) lim x->infinity ((3^x+4^x)/4)^(1/x)
2) lim x->infinity x^(3/2)((x+1)^(1/2)+(x-1)^(1/2)-2x^(1/2))

Please help me out...

 

[stapel]

\(\displaystyle \mbox{1) }\, \)\(\displaystyle \displaystyle \lim_{x\, \rightarrow\, \infty}\, \left(\dfrac{3^x\, +\, 4^x}{4}\right)^{\dfrac{1}{x}}\)

\(\displaystyle \mbox{2) }\,\)\(\displaystyle \displaystyle \lim_{x\, \rightarrow\, \infty}\, \left(x^{\frac{3}{2}}\right)\, \left((x\, +\, 1)^{\frac{1}{2}}\, +\, (x\, -\, 1)^{\frac{1}{2}}\, -\, 2x^{\frac{1}{2}}\right)\)

What are your thoughts? What methods have they given you? What have you tried? How far did you get? Where are you stuck?

Please be complete. Thank you!

 

[pka]

1) lim x->infinity ((3^x+4^x)/4)^(1/x)
2) lim x->infinity x^(3/2)((x+1)^(1/2)+(x-1)^(1/2)-2x^(1/2))

There are two useful theorems to use here:
If \(\displaystyle 0<c \) then \(\displaystyle (\sqrt[n]c)\to 1.\)
If \(\displaystyle 0<c<d\) then \(\displaystyle (\sqrt[n]{c^n+d^n})\to d\).