1.

Evaluate `lim_(x to 0) ((sqrt(1+x^(3))-sqrt(1-x^(3)))/(x^(2)))`

Answer» Given, `underset(xto1)"lim"(x^(7)-2x^(5)+1)/(x^(3)-3x^(2)+2)`
`=(x^(7)-2x^(5)+1)/(x^(3)-3x^(2)+2)` using `0/0` form
`=underset(xto1)"lim"(x^(7)-x^(5)-x^(5)+1)/(x^(3)-x^(2)-2x^(2)+2)`
`=underset(xto1)"lim"(x^(5)(x^(2)-1)-1(x^(5)-1))/(x^(2)(x-1)-2(x^(2)-1))`
On dividing numerator and denominator by (x-1), then
`=underset(xto1)"lim"((x^(5)(x^(2)-1))/(x-1)-(1(x^(2)-1))/(x-1))/((x^(2)(x-1))/(x-1)-(2(x^(2)-1))/(x-1))`
`=(underset(xto1)"lim"x^(5)(x+1)-underset(xto1)"lim"(x^(5)-1)/(x-1))/(underset(cto1)"lim"x^(2)-underset(xto1)"lim"(x+1))`
`=(1 xx 2-5xx(1)^(4))/(1-2xx2) = (2-5)/(1-4)`
`-3/-3=1`


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