Question

What is the limit of(1+(1/x))^x as x approaches infinity?

Answer 1

Make the limit of (1+(1/x))^x as x approaches infinity equal to any variable e.g. y, k. and take the natural logarithm of both sides.

Explanation:

`y=lim_(x-oo)(1+(1/x))^x`
`ln y =lim_(x-oo)ln (1+(1/x))^x`
`ln y =lim_(x-oo)x ln (1+(1/x))`
`ln y =lim_(x-oo) ln (1+(1/x))/x^-1`
if x is substituted directly, the value will be undefined, so
l'hopital's rule is applied.
l'hopital's rule says that if `lim_(x-a) f(x) =0= lim_(x-a) g(x)`,
then `lim_(x-a)(f(x)/g(x)) = lim_(x-a) ((f'(x))/(g'(x)))`
`ln y =lim_(x-oo)((1/(1+(1/x)))(0-1x^-2))/(-1x^-2)`
`ln y =lim_(x-oo)(1/(1+(1/x)))`
substitute for x
`ln y = (1/(1+0))`
`ln y = 1`
introduce exponential `e`
`e^ln y = e^1`
`y = e`
`y = e = lim_(x-oo)(1+(1/x))^x`
`lim_(x-oo)(1+(1/x))^x = e`