- #1
tomboi03
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For -1 < x < 1, x[tex]\neq[/tex]0 let
f(x)= [tex]\frac{\sqrt{1+x}-1}{x}[/tex]
i. Prove that
lim f(x)
x[tex]\rightarrow[/tex]0
exists and find it
(There is an easy proof and you get no credit for applying "hospital's rule")
Part (i) shows that f can be continued to a continuous function on (-1,1) if we assign this limit to be f(0) (this is assumed in subseguent parts)
ii. Show that the limit
lim 1/x [tex]\int[/tex] f(t) dt
x[tex]\rightarrow[/tex]0
exists
iii. Determine constants a0, a1, a2, a3 so that
[tex]\int[/tex] f(t) dt = a0 + a1x + a2x2 + a3x3 + x3 [tex]\rho[/tex](x)
(this integral goes from... 0 to x i couldn't figure out a way to put it on the integral.)
where
lim [tex]\rho[/tex](x) =0
x[tex]\rightarrow[/tex]0
I don't know how to go about this at all.
Please help me out
Thanks
f(x)= [tex]\frac{\sqrt{1+x}-1}{x}[/tex]
i. Prove that
lim f(x)
x[tex]\rightarrow[/tex]0
exists and find it
(There is an easy proof and you get no credit for applying "hospital's rule")
Part (i) shows that f can be continued to a continuous function on (-1,1) if we assign this limit to be f(0) (this is assumed in subseguent parts)
ii. Show that the limit
lim 1/x [tex]\int[/tex] f(t) dt
x[tex]\rightarrow[/tex]0
exists
iii. Determine constants a0, a1, a2, a3 so that
[tex]\int[/tex] f(t) dt = a0 + a1x + a2x2 + a3x3 + x3 [tex]\rho[/tex](x)
(this integral goes from... 0 to x i couldn't figure out a way to put it on the integral.)
where
lim [tex]\rho[/tex](x) =0
x[tex]\rightarrow[/tex]0
I don't know how to go about this at all.
Please help me out
Thanks