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1-Calculeu els límits següents: a) `lim (1+1/(5n))^(n/2)`
`lim ((1+1/(5n))^(5n))^(1/(5n)·n/2)` `lim ((1+1/(5n))^(5n))^(n/(10n))= e^(1/10)` b) `lim(1+1/n^2)^n`
`lim(1+1/n^2)^(n^2/n)` `lim((1+1/n^2)^(n^2))^(1/n)= e^0=1` c) `lim(1+1/n)^(n^2)`
`lim((1+1/n)^n)^n= e^n=e^(+\infty)=+\infty` d) `lim(1+3/(n+5))^n`
`lim(1+1/((n+5)/3))^(n·(n+5)/3·3/(n+5))` `lim(1+1/((n+5)/3))^((n+5)/3·n·3/(n+5))` `lim(1+1/((n+5)/3))^((n+5)/3·(3n)/(n+5))` `lim((1+1/((n+5)/3))^((n+5)/3))^((3n)/(n+5))=e^3` e) `lim(1+1/(2n))^(3n+1)`
`lim((1+1/(2n))^(2n))^((3n+1)/(2n))=e^(3/2)` f) `lim ((n+2)/(n+1))^n`
`lim (1+(n+2)/(n+1)-(n+1)/(n+1))^n` `lim (1+((n+2)-(n+1))/(n+1))^n` `lim (1+1/(n+1))^n` `lim (1+1/(n+1))^((n+1)·n/(n+1))` `lim ((1+1/(n+1))^(n+1))^(n/(n+1))=e^1=e` g) `lim((2n+3)/(2n-1))^((n^2+1)/n)`
`lim(1+(2n+3)/(2n-1)-(2n-1)/(2n-1))^((n^2+1)/n)` `lim(1+((2n+3)-(2n-1))/(2n-1))^((n^2+1)/n)` `lim(1+4/(2n-1))^((n^2+1)/n)` `lim(1+1/((2n-1)/4))^((n^2+1)/n)` `lim(1+1/((2n-1)/4))^((2n-1)/4·(n^2+1)/n·4/(2n-1))` `lim((1+1/((2n-1)/4))^((2n-1)/4))^((n^2+1)/n·4/(2n-1))` `lim((1+1/((2n-1)/4))^((2n-1)/4))^((4n^2+4)/(2n^2-n))=e^(4/2)=e^2` h) `lim(2-(n+3)/(n+1))^(3n)`
`lim(1+(n+1)/(n+1)-(n+3)/(n+1))^(3n)` `lim(1+((n+1)-(n+3))/(n+1))^(3n)` `lim(1+(-2)/(n+1))^(3n)` `lim(1+1/((n+1)/(-2)))^(3n)` `lim(1+1/((n+1)/(-2)))^((n+1)/(-2)·(3n)·(-2)/(n+1)` `lim((1+1/((n+1)/(-2)))^((n+1)/(-2)))^((-6n)/(n+1))=e^(-6/1)=e^(-6)=1/(e^6)` i) `lim (4-(3n)/(n+5))^(n^2)`
`lim (1+(3n+15)/(n+5)-(3n)/(n+5))^(n^2)` `lim (1+15/(n+5))^(n^2)` `lim (1+1/((n+5)/15))^(n^2)` `lim (1+1/((n+5)/15))^((n+5)/15·n^2·15/(n+5))` `lim ((1+1/((n+5)/15))^((n+5)/15))^((15n^2)/(n+5))=e^+\infty=+\infty` j) `lim(1-6/n)^(2n)`
`lim(1+1/(-n/6))^((-n/6)·(2n)·(-6/n))` `lim((1+1/(-n/6))^(-n/6))^((-12n)/n)=e^(-12)=1/(e^12)` k) `lim((n^2+5)/(n^2+1))^(5n)`
`lim(1+(n^2+5)/(n^2+1)-(n^2+1)/(n^2+1))^(5n)` `lim(1+4/(n^2+1))^(5n)` `lim(1+1/((n^2+1)/4))^(5n)` `lim(1+1/((n^2+1)/4))^(((n^2+1)/4)·(5n)·(4/(n^2+1)))` `lim((1+1/((n^2+1)/4))^((n^2+1)/4))^((20n)/(n^2+1))=e^0=1` l) `lim((n^3-3)/(2n^2-6))^(n^3)`
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