Answer
a) $B$
b) $A$
c) $A$
d) $A$
Work Step by Step
a). Step 1. Given $\lim_{x\to0^+}f(x)=A$ and $\lim_{x\to0^-}f(x)=B$, we need to identify the new limit based on to which value the new $x$ approaches.
Step 2. For $\lim_{x\to0^+}f(x^3-x)$, since $x\to 0+$, we have $x\gt0$ and $x^2-1\lt0$, thus $x^3-x=x(x^2-1)\lt0$
Step 3. Letting $y=x^3-x$, we have $x\to 0^+, y\to 0^-$, thus $\lim_{y\to0^-}f(y)=B$ which gives $\lim_{x\to0^+}f(x^3-x)=B$
b) Similarly, when $x\to0^-, x\lt0, y\to 0^+$, thus $\lim_{y\to0^+}f(y)=A$, which gives $\lim_{x\to0^-}f(x^3-x)=A$
c) Let $z=x^2-x^4=x^2(1-x^2)$, we have when $x\to0^+, z\to0^+$, thus $\lim_{z\to0^+}f(z)=A$ which gives $\lim_{x\to0^+}f(x^2-x^4)=A$
d) Similarly, when $x\to0^-, z\to0^+$, thus $\lim_{z\to0^+}f(z)=A$, which gives $\lim_{x\to0^-}f(x^2-x^4)=A$
