The solution goes like this
From the given lim_(x rarr oo) x^sin (1/x)
We test first. lim_(x rarr oo) x^sin (1/x) this results to oo^0
We apply logarithms
Let y=x^sin (1/x)
Take the logarithm of both sides
y=x^sin (1/x)
ln y=ln x^sin (1/x)
ln y=sin (1/x)*ln x
ln y=sin (1/x)*ln x
Test the limit of ln y and it takes the form 0*oo
We arrange it so that it will be oo/oo
lim_(x rarr oo) ln y=lim_(x rarr oo) (ln x)/(1/sin (1/x))=oo/oo
Use continuous derivatives on both numerator and denominator until we come up with a real number.
lim_(x rarr oo) ln y=lim_(x rarr oo) (1/x)/((sin (1/x)*0-1*cos(1/x)(-1/x^2))/(sin^2 (1/x)))
lim_(x rarr oo) ln y=lim_(x rarr oo) (1/x)/((1*cos(1/x)(1/x^2))/(sin^2 (1/x)))=lim_(x rarr oo) (sin^2 (1/x))/(1/x*cos(1/x))=0/(0*1)=0/0
lim_(x rarr oo) ln y=0/0
Differentiate again
lim_(x rarr oo) ln y=
=lim_(x rarr oo) (2sin (1/x)cos (1/x)*(-1/x^2))/(1/x(-sin (1/x)(-1/x^2))+cos (1/x)(-1/x^2))
=lim_(x rarr oo) (2sin (1/x)cos (1/x))/(-1/x*sin (1/x)+cos (1/x))=(2*0*1)/(0+1)=0/1=0
We now have
lim_(x rarr oo) ln y=0
Finally,
lim_(x rarr oo) y=e^0=1
And therefore
color(red)(lim_(x rarr oo) x^sin (1/x)=1)
God bless....I hope the explanation is useful.
