Vtyjkyuk

I'm working through problems in a real analysis textbook. I don't have any solutions. I am using the following definition for a limit:

Definition: Given a function $f : D rightarrow mathbbR$ and a limit point $x_0 in D$, for a number $ell$, we write

$$lim_xto x_0 f(x) = ell $$

provided that whenever $x_n in D - x_0$ that converges to $x_0$,

$$lim_ntoinfty f(x_n) = ell.$$

Problem:

I am trying to show that the limit

$$lim_xto 0 dfrac1 + 1/x^21 + 1/x$$

does not exist.

My attempt:

Let $x_n$ be a sequence in $D - 0$ that converges to $0$. Now we need to show that

$$lim_nto infty dfrac1 + 1/x_n^21 + 1/x_n = ell.$$

Rewrite the limit as follows:

$$ lim_nto inftydfrac1 + 1/x_n^21 + 1/x_n = lim_nto infty dfracx_n^2 + 1x_n^2 + x_n = dfrac10,$$

which is undefined. Thus, we can conclude that our limit does not exist.

As I know, in order to show in this way the limit doesn't exist we
have to address to a specific sequence. Take for example $a_n=frac1n$
which converges to $0$, but

beginalign*
fleft(a_nright) & =fleft(frac1nright)=frac1+frac1left(frac1nright)^21+frac1left(frac1nright)=frac1+n^21+n=fracleft(1+nright)^2-2n1+n=1+n-frac2n1+n=\
& =1+n-frac2left(1+nright)-21+n=-1+n+frac21+ntoinfty
endalign*

The sequence $x_n=1/n$ converges to $0$; on the other hand,
$$
f(1/n)=frac1+n^21+n>fracn^22n=fracn2
$$

Therefore, the sequence $f(1/n)$ doesn't converge.