<s>[INST] Find the limit given that $f(1)=1$, $f(x+y)=f(x)+f(y)+2xy$ and $f\left(\frac{1}{x}\right)=\frac{f(x)}{x^4}$ Given the function $f: \mathbb{R}\to \mathbb{R}$ satisfies:
\begin{cases}
f(1)=1 \\
f(x+y)=f(x)+f(y)+2xy,\quad&\forall x,y \in \mathbb{R}\\
f\left(\frac{1}{x}\right)=\frac{f(x)}{x^4},&\forall x\neq0
\tag{1}\label{eqn1}
\end{cases}
Then find the limit of:
\begin{align}
L=\lim_{x \to 0} \frac{\frac{1}{e^{2f(x)}}-\sqrt[3]{1+f(x)}}{\ln\bigl(1+f(x)\bigr)}
\end{align}
My attempts
I can easily guess that $f(x)=x^2$, which totally satisfies \eqref{eqn1}. What I did was that I set $g(x)=f(x)-x^2$, then I got:
\begin{cases}
g(x)+g(y)=g(x+y)\\
g\left(\frac{1}{x}\right)x^4=g(x)
\tag{2}\label{eqn2}
\end{cases}
From \eqref{eqn1}, substitute $x=y=0$ we have $f(0)=0$
Now $g(1)=g(0)=0$, then I substitute $y=1-x$ in \eqref{eqn2} and got that
\begin{align}
g\left(\frac{1}{x}\right)x^4+g\left(\frac{1}{1-x}\right)(1-x)^4=0
\end{align}
Because the values of either $\frac1x$ or $\frac1{1-x}$ must lie between $0$ and $1$, I suspect that we can prove $g \equiv 0$.
Even if I have $f$ then, I'm still clueless about how to calculate the limit.
Any idea or solution or suggestion on the tools?
Any help is appreciated!
 [/INST] Calculating the given limit is standard, when you prove that $ f ( x ) = x ^ 2 $ for all $ x \in \mathbb R $. Let's focus on the nontrivial part of the problem. Using $ g ( 1 ) = 0 $,
$$ g ( x + y ) = g ( x ) + g ( y ) \text , \tag 0 \label 0 $$
and
$$ g \left ( \frac 1 x \right ) = \frac { g ( x ) } { x ^ 4 } \text , \tag 1 \label 1 $$
you can see that for any $ x \in \mathbb R \setminus \{ - 1 , 0 \} $,
\begin{align*}
\frac { g \left ( x ^ 2 \right ) } { x ^ 4 ( x + 1 ) ^ 4 } + \frac { g ( x ) } { x ^ 4 ( x + 1 ) ^ 4 }
& = \frac { g \left ( x ^ 2 + x \right ) } { \left ( x ^ 2 + x \right ) ^ 4 } \\
& \stackrel { \eqref {1} } = g \left ( \frac 1 { x ^ 2 + x } \right ) \\
& \stackrel { \eqref {0} } = g \left ( \frac 1 x \right ) - g \left ( \frac 1 { x + 1 } \right ) \\
& \stackrel { \eqref {1} } = \frac { g ( x ) } { x ^ 4 } - \frac { g ( x + 1 ) } { ( x + 1 ) ^ 4 } \\
& \stackrel { \eqref {0} } = \frac { g ( x ) } { x ^ 4 } - \frac { g ( x ) + g ( 1 ) } { ( x + 1 ) ^ 4 } \\
& = \frac { \left ( 4 x ^ 3 + 6 x ^ 2 + 4 x + 1 \right ) g ( x ) } { x ^ 4 ( x + 1 ) ^ 4 } \text ,
\end{align*}
which implies
$$ g \left ( x ^ 2 \right ) = 2 x ^ 5 ( x + 1 ) ^ 4 \left ( x ^ 2 + 3 x + 1 \right ) g ( x ) \text . \tag 2 \label 2 $$
Using \eqref{0} and \eqref{2} you get
\begin{align*}
& & & 2 x ^ 5 ( x + 1 ) ^ 4 \left ( x ^ 2 + 3 x + 1 \right ) g ( x ) + 2 g ( x y ) + 2 y ^ 5 ( y + 1 ) ^ 4 \left ( y ^ 2 + 3 y + 1 \right ) g ( y ) \\
& & \stackrel { \eqref {2} } = & g \left ( x ^ 2 \right ) + 2 g ( x y ) + g \left ( y ^ 2 \right ) \\
& & \stackrel { \eqref {0} } = & g \left ( ( x + y ) ^ 2 \right ) \\
& & \stackrel { \eqref {2} } = & 2 ( x + y ) ^ 5 ( x + y + 1 ) ^ 4 \left ( ( x + y ) ^ 2 + 3 ( x + y ) + 1 \right ) g ( x + y ) \\
& & \stackrel { \eqref {0} } = & 2 ( x + y ) ^ 5 ( x + y + 1 ) ^ 4 \left ( ( x + y ) ^ 2 + 3 ( x + y ) + 1 \right ) \bigl ( g ( x ) + g ( y ) \bigr ) \text ,
\end{align*}
which for $ y = 1 $ shows that
$$ \Bigl ( ( x + 1 ) ^ 5 ( x + 2 ) ^ 4 \left ( x ^ 2 + 5 x + 5 \right ) - x ^ 5 ( x + 1 ) ^ 4 \left ( x ^ 2 + 3 x + 1 \right ) - 1 \Bigr ) g ( x ) = 0 \text . \tag 3 \label 3 $$
Therefore, $ g ( x ) = 0 $ for all $ x \in \mathbb R \setminus A $, where $ A $ is a finite set of real numbers consisting of $ - 1 $, $ 0 $, and finitely many roots of the polynomial appearing on the left-hand side of \eqref{3}. For any $ a \in A $, choosing a positive integer $ n $ large enough so that $ a + n > b $ for all $ b \in A $, you can use \eqref{0} to see that $ g ( a ) = 0 $, and hence $ g ( x ) = 0 $ for all $ x \in \mathbb R $, or equivalently $ f ( x ) = x ^ 2 $ for all $ x \in \mathbb R $.
</s>