Evaluate `lim_(xto0^(+)) (1)/(x)cos^(-1)((sinx)/(x))`.

1.	Evaluate `lim_(xto0^(+)) (1)/(x)cos^(-1)((sinx)/(x))`.
Answer» `underset(xto0^(+))lim(1)/(x)cos^(-1)((sinx)/(x))` `=underset(xto0^(+))lim(sin^(-1)(sqrt(1-(sin^(2)x)/(x^(2)))))/(x)` `=underset(xto0^(+))lim(sin^(-1)(sqrt(1-(sin^(2)x)/(x^(2))))sqrt(1-(sin^(2)x)/(x^(2))))/sqrt(1-(sin^(2)x)/(x^(2)))` `=underset(xto0^(+))lim(sqrt(x^(2)-sin^(2)x))/(x^(2))` `=underset(xto0^(+))limsqrt((x+sinx)/(x).(x-sinx)/(x^(3)))` `=sqrt(2)underset(xto0^(+))limsqrt((x-(x-(x^(3))/(3!)))/(x^(3)))` `=sqrt(2)sqrt((1)/(6))` `=sqrt((1)/(3))`

Discussion

`lim_(xto oo) (1)/(n) {1+e^(1//n)+e^(2//n)+e^(3//n)+.....+e^((n-1)/(n))}` is equal toA. eB. `-e`C. `e-1`D. `1-e`
`lim_(xto oo) ((nsqrt(a)+nsqrt(b))/(2))^n,a,b,gt 0` equalsA. 1B. `sqrt(pq)`C. `pq`D. `(pq)/(2)`
If `lim_(xto oo){(x^2+1)/(x+1)-(ax+b)}to oo`, thenA. `a in (1,oo)`B. `a ne 1 , b in R`C. `a in (-oo,1)`D. none of these
Evaluate:\(\lim\limits_{x \to 1}\frac{\sqrt {1+x}-\sqrt {1-x}}{1+x}\)
\(f\left(x\right)=\left[\left(\frac{min\left(t^2+4t+6\right)\sin \left(x\right)}{x}\right)\right]\)where [.] represents greatest integer function, then find \(\lim _{x\to 0}\left(f\left(x\right)\right)\)
`lim_(xrarr0) ((1-cos2x)^2)/(2xtanx-xtan2x)` , isA. 2B. `-(1)/(2)`C. `(1)/(2)`D. `-2`
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`lim_(xrarroo) ((x+2)/(x+1))^(x+3)` is equal toA. 1B. eC. `e^2`D. `e^3`
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