Help with this limit...?

lim (x->0+) (x^(a) log(x))/(log(1 + x^2) (sin(x^2) - sin^2(x)))xalog(x)log(1+x2)(sin(x2)sin2(x)) a>0

1 Answer
Jul 28, 2018

lim_(x to 0^+)(x^alog(x))/(log(1+x^2)(sin(x^2)-sin^2(x)))=0

Explanation:

lim_(x to 0^+)(x^alog(x))/(log(1+x^2)(sin(x^2)-sin^2(x))), a>0

=x^a/log(1+x^2)*log(x)/(sin(x^2)-sin^2(x))

Now let observe x^a/log(1+x^2)
The limit is 0/0, and both functions are continuous on 0^+.

Using L'Hôpital's therorem
=(ax^(a-1))/((2x)/(1+x^2))

=a/2(x^(a-2)+x^(a-1))

And lim_(x to 0^+) a/2(x^(a-2)+x^(a-1))=0
Now we have lim_(x to 0^+) ((a/2(x^(a-2)+x^(a-1)))log(x))/(sin(x^2)-sin^2(x))

Now Let's take a look at (a/2(x^(a-2)+x^(a-1)))/(sin(x^2)-sin^2(x))

Using L'Hôpital's theorem

=a/2((a-2)x^(a-3)+(a-1)x^(a-2))/(2xcos(x^2)-2cos(x)sin(x))
Using L'Hôpital again
f=a/4((a-2)(a-3)x^(a-4)+(a-1)(a-2)x^(a-3))/(cos(x^2)+2x²sin(x²)-sin(2x))
With lim_(x to 0^+)f(x) =0
Finally, lim_(x to 0^+)f(x)log(x)=lim_(x to 0^+)log(x^(f(x)))=log(1)=0
\0/ Here's our answer !