How can i prove that limit [closed]
Solution 1:
The proposition is wrong. I'll give an very elementary counter-example.
I'll let $f_j(x)=x+j^2x^2$, then it satisfies $\lim_{x\to0^+}f_j(x)=0$ and $\lim_{x\to0^+}\frac{f_j(x)}x=1$, and $f_j$ strictly inctrases when $x>0$.
But $f_4(x)-f_3(x)=7x^2$ and $f_2(x)-f_1(x)=3x^2$, so $\lim_{x \rightarrow 0^{+}}\frac{f_4(x) - f_3(x)}{f_2(x) - f_1(x)} = \frac73 $.