Real analysis elementary limit property

Question

I'm reading through some lecture notes to prepare myself for analysis next semester and stumbled along the following exercises:

a) Prove that $\lim_{x\to0} f(x)=b$ is equivalent to the statement $\lim_{x\to0} f(x^3)=b$.

b) Give an example of a map where $\lim_{x\to0} f(x^2)$ exists, but $\lim_{x\to0} f(x)$ does not.

for b) I was thinking about the following piecewise function:

$f(x)=\begin{cases} -1 & x < 0 \\ 1 & x \geq0 \end{cases}$

is this a good example?

for (a), I don't have any concrete tools to work with, I can't write down any explicit $\epsilon$ or $\delta$, so what can I do?

Answer 1

Suppose, for every $\epsilon>0$, whenever $|x-0|< \delta$, it holds true that $|f(x)-b|<\epsilon$. With this in mind we can make the case for $x^3$,

By using the fact that $x^3$ is bijective and

for any $x \in \mathbb{R}$, $p(x)$ is true$\iff$ for any $x^3 \in \mathbb{R}$, $p(x^3)$ is true

since $x^3$ tends to $0$, as $x$ tends to zero, we can replace the orginal limit by $\lim_{x^3\to 0} f(x^3)$, after substituting $x $ by $x^3$ to get:

for every $\epsilon>0$, whenever $|x^3-0|< \delta$, it holds true that $|f(x^3)-b|<\epsilon$. QED.

An alternative way would be via the substitution rule for limits:

Let $\lim_{y \to 0} f(y)=0$, and notice if we choose $\lim_{x \to 0}y= \lim_{x\to 0} x^3=0$ then by substitution $\lim_{x\to 0} f(x^3)=0$,

Answer 2

Yes your example for point b) is a good example.

For a) the property is true for continuity of the function $x^3$ and since $x^3 \to 0$ as $x\to 0$. Yes we can prove that by the $\epsilon-\delta$ definition.

Refer also to Formal basis for variable substitution in limits.

Answer 3

For a) You can use the following in an epsilon delta proof:

for any $x \in \mathbb{R}$, $p(x) \iff$ for any $x^3 \in \mathbb{R}$, $p(x^3)$