求過程大學高數真的難？

01-15

求例題二過程怎麼來的怎麼算

$[egin{gathered} w = mathop {lim }limits_{x o infty } {left( {frac{1}{x} + {2^{frac{1}{x}}}} ight)^x} hfill \ u = frac{1}{x}, hfill \ w = mathop {lim }limits_{u o 0} {left( {u + {2^u}} ight)^{frac{1}{u}}} = {e^{mathop {lim }limits_{u o 0} frac{1}{u}ln left( {u + {2^u}} ight){ ext{ - - - - - - - - - - - - - - - - - }}frac{0}{0}}} hfill \ = {e^{mathop {lim }limits_{u o 0} frac{{1 + {2^u}ln 2}}{{u + {2^u}}}}} = {e^{1 + ln 2}} = e{e^{ln 2}} hfill \ left( {z = {e^{ln 2}} Rightarrow ln z = ln 2 Rightarrow z = 2 = {e^{ln 2}}} ight) hfill \ = 2e hfill \ end{gathered} ]$

$[egin{gathered} w = mathop {lim }limits_{x o infty } {left( {frac{1}{x} + {2^{frac{1}{x}}}} ight)^x} hfill \ u = frac{1}{x}, hfill \ w = mathop {lim }limits_{u o 0} {left( {u + {2^u}} ight)^{frac{1}{u}}}, hfill \ w = mathop {lim }limits_{x o infty } 2{left[ {{{left( {1 + frac{1}{x}{2^{ - frac{1}{x}}}} ight)}^{x cdot {2^{frac{1}{x}}}}}} ight]^{{2^{ - frac{1}{x}}}}} hfill \ = mathop {lim 2}limits_{u o 0} {left[ {{{left( {1 + u{2^{ - u}}} ight)}^{frac{1}{u} cdot {2^u}}}} ight]^{{2^{ - u}}}} hfill \ = mathop {lim 2}limits_{u o 0} {left( {1 + u{2^{ - u}}} ight)^{frac{1}{u}}} hfill \ = mathop {lim 2}limits_{u o 0} {left( {1 + frac{u}{{{2^u}}}} ight)^{frac{1}{u}}} hfill \ = mathop {lim 2}limits_{u o 0} {left( {frac{{{2^u} + u}}{{{2^u}}}} ight)^{frac{1}{u}}} hfill \ = mathop {lim 2}limits_{u o 0} frac{{{{left( {{2^u} + u} ight)}^{frac{1}{u}}}}}{2} hfill \ = mathop {lim }limits_{u o 0} {left( {u + {2^u}} ight)^{frac{1}{u}}} hfill \ end{gathered} ]$

求過程 大學高數真的難？

求過程大學高數真的難？