1.

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

Answer» Here, we will use,
`lim_(x->a) (x^n-a^n)/(x-a) = na^(n-1)`

(i)`lim_(x->1)(x^15-1)/(x^10-1)``= lim_(x->1)((x^15-1^15)/(x-1))/((x^10-1^10)/(x-1))`
`=(15(1)^14)/(10(1)^9) = 15/10 = 3/2`

(ii)`lim_(x->0) (sqrt(1+x) -1)/x`
`= lim_(1+x->1)((1+x)^(1/2) - (1)^(1/2))/((1+x)-1)`
`=1/2(1)^(-1/2) = 1/2`


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