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x->0+
2^x = 1+(ln2)x + o(x)
3^x = 1+(ln3)x + o(x)
(2^x+3^x)/2 =1 +(1/2)(ln6)x +o(x)
lim(x->0+) [(2^x+3^x)/2]^(1/x)
=lim(x->0+) [1 +(1/2)(ln6)x]^(1/x)
=e^[(1/2)ln6]
=√6
f(x)
=(1/x)sinax ; x<0
=3 ; x=0
=[(2^x+3^x)/2]^(1/x) +b ; x>0
f(0-)=lim(x->0-) f(x) = lim(x->0-) (1/x)sinax = a
f(0) =3
f(0-)=f(0)
=> a=3
f(0+)
=lim(x->0+) f(x)
= lim(x->0+) { [(2^x+3^x)/2]^(1/x) +b }
= √6 +b
f(0+) = f(0)
=>
b=3-√6
2^x = 1+(ln2)x + o(x)
3^x = 1+(ln3)x + o(x)
(2^x+3^x)/2 =1 +(1/2)(ln6)x +o(x)
lim(x->0+) [(2^x+3^x)/2]^(1/x)
=lim(x->0+) [1 +(1/2)(ln6)x]^(1/x)
=e^[(1/2)ln6]
=√6
f(x)
=(1/x)sinax ; x<0
=3 ; x=0
=[(2^x+3^x)/2]^(1/x) +b ; x>0
f(0-)=lim(x->0-) f(x) = lim(x->0-) (1/x)sinax = a
f(0) =3
f(0-)=f(0)
=> a=3
f(0+)
=lim(x->0+) f(x)
= lim(x->0+) { [(2^x+3^x)/2]^(1/x) +b }
= √6 +b
f(0+) = f(0)
=>
b=3-√6
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