For a function $f:[-1,1]\to R$, consider the following statements:

Statement 1: If

$\u2018\underset{n\to \mathrm{\infty}}{lim}f\left(\frac{1}{n}\right)=f(0)=\underset{n\to \mathrm{\infty}}{lim}f(-\frac{1}{n})\u2018,$

then $f$ is continuous at $x=0$

Statement 2: If $f$ is continuous at $x=0$, then

$\underset{n\to \mathrm{\infty}}{lim}f\left(\frac{1}{n}\right)=\underset{n\to \mathrm{\infty}}{lim}f(-\frac{1}{n})=\underset{n\to \mathrm{\infty}}{lim}f({e}^{\frac{1}{n}}-1)=f(0)$

Then which of the above statements is/are true.

My Attempt:

I feel that $\underset{n\to \mathrm{\infty}}{lim}f\left(\frac{1}{n}\right)=f(0)$ is same as $\underset{x\to 0}{lim}f(x)=f(0)$, so $f$ should be continuous at $x=0$. So statement 1 must be true.

In statement 2, since $f$ is continuous at $x=0$ we have $\underset{n\to \mathrm{\infty}}{lim}f\left(\frac{1}{n}\right)=f\left(\underset{n\to \mathrm{\infty}}{lim}\frac{1}{n}\right)=f(0)$. By same logic we can prove $\underset{n\to \mathrm{\infty}}{lim}f(-\frac{1}{n})=\underset{n\to \mathrm{\infty}}{lim}f({e}^{\frac{1}{n}}-1)=f(0)$

Can there be counter-examples to what I am thinking