Limit of f(x) knowing limit of f(x)/x

Given that: $$\lim_{x\to0} \frac{f(x)}{x}=0$$ How can I prove that: $$\lim_{x\to0} f(x)=0$$ ?

Would the L'Hospital's Rule be applicable?

6 Answers

Hint :

$f(x)=\frac{f(x)}{x}{x}$ and product rule.

If by contradiction $f(x)\to a\neq0,\;\infty$, your limit $\lim f(x)/x$ would be $\infty$. How can you obtain a contradiction if $\lim f(x)$ doesn't exist?

By definition, if $\lim_{x \to 0} f(x)/x = 0$ then for all $\epsilon>0$ there exist small enough $x$ such that

$$-\epsilon<f(x)/x<\epsilon$$ therefore, since $x\ne0$ $$-\epsilon\cdot x<f(x)<\epsilon\cdot x$$ and when $x\in (0, 1)\cup(-1, 0)$ we get $$-\epsilon<f(x)<\epsilon$$ which implies that $\lim_{x \to 0} f(x) = 0$

Because the $\lim_{x\to0} \frac{f(x)}{x}=0$, this means that $f(x)$ tends to $0$ faster than $x$, or that $|f(x)|\le x$.

Therefore:

$$0\le|f(x)|\le x $$

Using the squeeze theorem, and because $\lim_{x\to0} x=0$, $\lim_{x\to0} f(x){x}=0$.

By the definition of a limit, f (x) / x < eps for every eps if x is small enough. Take eps = 1, so f (x) / x < 1 if x < eps1, or f (x) < x if x < eps1. Take the definition of the limit again; f (x) < eps if you take x < min (eps, eps1).

Obviously you don't need that the limit of f (x) / x is 0. If the limit is c, then f (x) / x < c+1 for small x, so f (x) < x * (c + 1) for small x, and f (x) < eps if x < eps / (c + 1).

Obviously you'll have to add a few absolute values in the argument.

This is not a fancy proof. It is a grinding one from the epsilon-delta definition of limit suitable for an introductory calculus class.

The core of the idea is that if $f(x)$ is bounded away from $0$ as $x$ goes to zero, then $f(x)/x$ is also bounded away from zero, as for small $x$, dividing by $x$ makes things further from zero.

And things bounded away from zero don't have a limit of zero.

Suppose the limit as $x$ goes to zero of $f(x)$ is $a$, and $a > 0$.

Let $\epsilon = a/2$. Then there is a $\delta$ such that for all $0 < x < \delta$, $a/2 < f(x) < 3a/3$.

Let $\delta_0$ be the minimum of $1/2$ and $\delta$.

(statement 1): Then for $0 < x < \delta_0$, $f(x)/x \geq f(x)*2 \geq 2a/2 = a$

If the limit of $f(x)/x$ as $x$ goes to zero is $0$, then let $\epsilon = a/2$.

(statement 2): Then there exists a $\delta_1 > 0$ such that for all $0 < x < \delta_1$, $-a/2 < f(x)/x < a/2$

Let $\delta_2$ be the least of $\delta_0$ and $\delta_1$.

For $x < \delta_2 <= \delta_1$, $f(x)/x < a/2$ by (2).

For $x < \delta_2 <= \delta_1$, $f(x)/x \geq a$ by (1).

Thus $a/2 > a$ or $1/2 > 1$, a contradiction.

A similar result holds of $a < 0$. Spotting the places where I implicitly assumed $a$ was positive may be interesting.