If a and b are positive and [x] denotes greatest integer less than or equal to x, then find `lim_(xto0^(+)) x/a[(b)/(x)].`

If a and b are positive and [x] denotes greatest integer less than or

1.	If a and b are positive and [x] denotes greatest integer less than or equal to x, then find `lim_(xto0^(+)) x/a[(b)/(x)].`
Answer» `L=underset(xto0^(+))limx/a[(b)/(x)].` `=underset(xto0^(+))limx/a(b/x-{b/x})," "`where `{.}` represents the functional part function `=underset(xto0^(+))lim(b/a-x/a{b/x})` `=b/a-1/aunderset(xto0^(+))lim({b/x}}/(1/x)` Since `0lt{b/x}lt1 " and "underset(xto0^(+))lim1/x=oo,` we have `L= b/a-0=b/a`

`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 \(\lim _{x\to 0}\left(f\left(x\right)\right)\)
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`lim_(x->0)((1-cos2x)sin5x)/(x^2sin3x)`A. `(10)/(3)`B. `(3)/(10)`C. `(6)/(5)`D. `(5)/(6)`
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