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If f(x)=(tanx)/(x), then find lim(xto0)(...

If `f(x)=(tanx)/(x)`, then find `lim_(xto0)([f(x)]+x^(2))^((1)/({f(x)}))`, where `[.]` and `{.}` denotes greatest integer and fractional part function respectively.

Text Solution

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We know that `underset(xto0)lim(tanx)/(x)=1^(+)`
`:." "underset(xto0)lim[(tanx)/(x)]=1`
and `underset(xto0)lim{(tanx)/(x)}=underset(xto0)lim((tanx)/(x)-[(tanx)/(x)])`
`=underset(xto0)lim((tanx)/(x)-1)`
`:." "underset(xto0)lim([f(x)]+x^(2))^((1)/({f(s)}))=underset(xto0)lim(1+x^(2))^(underset(xto0)lim((1)/(tanx))/(x)-1)`
`=e^(underset(xto0)lim((x^(2))/(tanx))/(x)-1)`
`=e^(underset(xto0)lim(x^(3)cosx)/(sinx-xcosx))`
`=e^(underset(xto0)lim(x^(3))/((x+(x^(3))/(3!)+...)-x(1-(x^(2))/(2!)+...)))`
`=e^(underset(xto0)lim(x^(3))/(-(x^(3))/(3!)+(x^(3))/(2!)))=e^(3)`
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