Evaluate (i) `lim_(x""->1)(x^(15)-1)/(x^(10)-1)` (ii) `lim_(x""->0)(sqrt(1+x)-1)/x`

Explanation

(i) Given limit: `lim_(x""->1)(x^(15)-1)/(x^(10)-1)`
`lim_(x->1)((x^(15)-1)/(x-1))/((x^(10)-1)/(x-1))`
` lim_(x->1)((x^15-1^15)/(x-1))((x-1)/ (x^15-1^15))`
we know that `lim_(x->a)(x^n-a^n)/(x-a) = n*a^(n-1)`
so, using this formula
` lim_(x->1)((x^15-1^15)/(x-1))((x-1)/ (x^15-1^15))`
`=15/10`
`=3/2`

(ii)Given limit: `lim_(x""->0)(sqrt(1+x)-1)/x`
by applying Rationalization
`lim_(x""->0)(sqrt(1+x)-1)/x times (sqrt(1+x)+1)/ (sqrt(1+x)+1)`
>`lim_(x""->0)((1+x)-1)/x(sqrt(1+x)+1)`
`lim_(x->0)1/(sqrt(1+x)+1)`
`=1/2`