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Evaluate lim(xto1) (x^(4)-sqrt(x))/(sqrt...

Evaluate `lim_(xto1) (x^(4)-sqrt(x))/(sqrt(x)-1)`.

Text Solution

Verified by Experts

Given, `underset(xto1)"lim"(x^(4)-sqrt(x))/(sqrt(x)-1) = underset(xto1)"lim"(sqrt(x)[(sqrt(x))^(7//2)-1])/(sqrt(x)-1)`
`=underset(xto1)"lim"((x)^(7//2)-1)/(sqrt(x)-1).underset(xto1)"lim"sqrt(x)` `[therefore underset(xtoa)"lim"f(x).g(x)= underset(xtoa)"lim"f(x). underset(xtoa)"lim"g(x)]`
`=underset(xto1)"lim"((x^(7//2)-1)/(x-1))/(((x)^(1//2)-1)/(x-1)).1`
`=(underset(xto1)"lim"(x^(7//2)-1)/(x-1))/(underset(xto1)"lim"(x^(1//2)-1)/(x-1))` `[therefore underset(xtog)"lim"(f(x))/(g(x))= (underset(xtoa)"lim"f(x))/(underset(xtoa)"lim"g(x)]]`
`=(7/2(1)^(7/2-1))/(1/2(1)^(1/2)-1)=(7/2)/(1/2)=7`
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