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## Homework Statement

:Find [/B]

lim

_{x->∞}(x

^{α}(sin

^{2}x!)/(x+1)

α∈(0,1)

Options are:

a)0

b)1

c)inifinity

d)does not exist

## Homework Equations

: -[/B]## The Attempt at a Solution

:lim

_{x->∞}(x

^{α}sin

^{2}x!)/(x+1)[/B]

Dividing the numerator and denominator by x

^{α},

we have:lim

_{x->∞}sin

^{2}x!)/(x

^{1-α}+x

^{-α})

clearly x

^{−α}is tending to zero as x tends to infinity

and thus we have 1/x

^{1-α}tending to zero

[as x tends to infimity, we can say the denomimator tends to x

^{1-α}]

thus we have (a number tending to zero)*(a sinusoidal function which is largely changing values as x tends to infinity)

so i feel limit does not exist because of that oscillating part.

My friends feel answer is zero as the sin part has a finite value, so we have

(tending to zero)*(finite number) which gives tending to zero.

And THE ANSWER GIVEN IN TEXT IS INFINITY.

Help please.

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