f) [latex]lim_{n o infty}left(sqrt{ n+sqrt{n+sqrt{n}} } - sqrt{n} ight) =[/latex] [latex]lim_{n o infty} frac{left(sqrt{ n+sqrt{n+sqrt{n}} } - sqrt{n} ight)left(sqrt{ n+sqrt{n+sqrt{n}} } +sqrt{n} ight) }{sqrt{ n+sqrt{n+sqrt{n}} } + sqrt{n}} =[/latex] [latex]lim_{n o infty} frac{ n+sqrt{n+sqrt{n}} -n}{sqrt{ n+sqrt{n+sqrt{n}} } + sqrt{n}} =[/latex] [latex]lim_{n o infty} frac{left(sqrt{n+sqrt{n}} ight): sqrt{n} }{left(sqrt{ n+sqrt{n+sqrt{n}} } + sqrt{n} ight) : sqrt{n} }=[/latex] [latex]lim_{n o infty} frac{ sqrt{ frac{n+sqrt{n}}{n} } }{sqrt{ frac{n+sqrt{n+sqrt{n}}}{n} } +1}=[/latex] [latex]lim_{n o infty} frac{ sqrt{1+ sqrt{ frac{n}{n^2} } } }{ sqrt{1+ sqrt{ frac{n+ sqrt{n} }{n^2} } } +1}=[/latex] [latex]lim_{n o infty} frac{ sqrt{1+ sqrt{ frac{1}{n} } } }{ sqrt{1+ sqrt{ frac{n+ sqrt{n} }{n^2} } } +1}=[/latex] [latex]lim_{n o infty} frac{ sqrt{1+ sqrt{ frac{1}{n} } } }{ sqrt{1+ sqrt{ frac{n}{n^2} + frac{ sqrt{n} }{n^2} }}+1 }=[/latex] [latex]lim_{n o infty} frac{ sqrt{1+ sqrt{ frac{1}{n} } } }{ sqrt{1+ sqrt{ frac{1}{n} + sqrt{ frac{n}{n^4} } }} +1 }=[/latex] [latex]lim_{n o infty} frac{ sqrt{1+ sqrt{ frac{1}{n} } } }{ sqrt{1+ sqrt{ frac{1}{n} + sqrt{ frac{1}{n^3} } }} +1 }=[/latex] [latex]frac{ sqrt{1+ sqrt{0} } }{ sqrt{1+ sqrt{0+ sqrt{ 0} }} +1 }=[/latex] [latex]frac{ sqrt{1+0 } }{ sqrt{1+ sqrt{0+ 0}} +1 }=[/latex] [latex]frac{ sqrt{1} }{ sqrt{1} +1 }= frac{1}{1+1}= frac{1}{2}[/latex]
