Question

Help with this limit...?

lim (x->0+) `(x^(a) log(x))/(log(1 + x^2) (sin(x^2) - sin^2(x)))` a>0

Answer 1

`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 !