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Prove or disprove given statements for functions such that f(x) = o(g(x))

Let and be functions, both differentiable in a neighborhood of 0, with and such that

Prove or disprove each the following statements.

1. as .
2. as .

1. True.
Proof. Since as we know by the definition of that

Thus, for every there exists a such that

So, for we have

The final line follows since by hypothesis. Therefore,

Hence,

By definition, we then have

2. False.
Consider for and for . Then, for ,

For we have .

Next,

Since we have as . However, since

does not exist.

I think part (b) could use a more simpler counterexample. For example, let $f(x)=x$ and $g(x)=1$ defined for all real numbers. Then $f$ and $g$ both have derivatives in some interval containing 0, and clearly $g$ is always positive, also $f(x)=o(g(x))$ as $x\rightarrow 0$. But we can see that $1=f^{\prime}(x)=o(g^{\prime}(x))=o(0)$ is not true, since o(0) is undefined.