Show that if $f : (a,\infty)\to \mathbb R$ is such that $lim_{x\to \infty}xf(x)=L$ where $ L \in \mathbb R$, then $lim_{x\to \infty}f(x)=0$

Question

Show that if $f : (a,\infty)\to \mathbb R$ is such that $\lim_{x\to \infty}xf(x)=L$ where $ L \in \mathbb R$, then $\lim_{x\to \infty}f(x)=0$

Let $\epsilon>0$,$\exists K(\epsilon)>0$: $\forall x:|x|>K(\epsilon)\implies \left|\left|xf(x)\right|-\left|L\right|\right| \leq|xf(x)-L|<\epsilon$

$\epsilon>0$,$\exists K(\epsilon)>0$: $\forall x:|x|>K(\epsilon)\implies \frac{|L|-\epsilon}{|x|}<|f(x)|<\frac{|L|+\epsilon}{|x|}$

Choose $M=\max\left\{\frac{|L|+\epsilon}{\epsilon}, \, K(\epsilon)\right\}\implies \forall x>M ,|f(x)|<\epsilon$

Am I correct? Can I choose $M$ like this?

Answer 1

Let $x\to \infty$ in

$$\frac{L-\epsilon}{x}<f(x)<\frac{L+\epsilon}{x}$$

and apply the squeeze theorem.

Answer 2

1) Let $\epsilon >0$ be given.

Need to show:

There is a $K$, real, positive, such that

$x>K$ implies $|f(x)| < \epsilon$.

2) Given:

Let $\varepsilon >0$ be given.

There is a $M$, real, positive, such that

$x \gt M$ implies $|xf(x)-L| < \varepsilon.$

3) There is a $N$, real, positive,

such $x >N$ implies

$|1/x| \lt \min (1/|L|, 1/\varepsilon) \epsilon/2.$

Let $K= \max(M,N)$.

Then for $x \gt K$:

$|f(x)| = |1/x||xf(x)|=$

$|1/x| |xf(x)-L+L| \le$

$|1/x||xf(x)-L| +|1/x||L| \lt$

$|1/x|\varepsilon +|1/x||L| \lt \epsilon/2 +\epsilon/2 =\epsilon$.