Question

Question #e9dcc

Answer 1

`5/6`

Explanation:

Since direct substitution gives `0/0`, we can use L'Hopital's Rule:

`lim_(x to 0)(sqrt(1+x)-root[3]{1-x})/x =`
`lim_(x to 0)(d/dx(sqrt(1+x)-root[3]{1-x}))/(d/dx(x)) =`
`lim_(x to 0)(1/(2sqrt(1+x))+(1/3)(1-x)^(-2/3))/1 =`

now direct substitution gives:

`lim_(x to 0)(1/(2sqrt(1+x))+(1/3)(1-x)^(-2/3))/1 = 1/2+1/3 = 5/6`