In my high school Limit of a function was introduced as below.
We say $\lim_{x\to a^-} f(x)$ is the expected value of $f$ at $x = a$ given the values of $f$ near $x$ to the left of $a$. This value is called the left hand limit of $f$ at $a.$ We say $\lim_{x\to a^+} f(x)$ is the expected value of $f$ at $x = a$ given the values of $f$ near $x$ to the right of $a$. This value is called the right hand limit of $f$ at $a.$ If the right and left hand limits coincide, we call that common value as the limit of $f(x)$ at $x=a$ and denote it by $\lim_{x\to a} f(x)$.
For initial examples we first compute the value of $f$ at points very close to $a$ and then deduce the limit. Then move to polynomials. Then if functions under consideration are rational functions. We first evaluate these functions at the prescribed points. If this is of the form $\frac{0}{0}$ , we try to rewrite the function cancelling the factors which are causing the limit to be of the form $\frac{0}{0}$ .
Now in undergraduate studies we were introduced to the $\epsilon- \delta$ definition of limit. I understand it fully. I just needed an example of a quick example showcasing how the high school definition of limit is incomplete and is not formal enough. An example that illustrates why we need formalisation of the phenomenon that as $x$ approaches $a$, $f(x)$ approaches its limit. Can anyone please provide such an example?
This may be the kind of example you want. Suppose $f(x)=x$ if $x$ is rational, $f(x)=17$ if $x$ is irrational, and you want to determine whether $\lim_{x\to0}f(x)$ exists. So, you look at what happens at values of $x$ near zero. Well, if you make life easy on yourself and only look at rational values of $x$ near zero, you'll conclude the limit exists, and equals zero. If your best friend is more imaginative, and only looks at irrational values of $x$ near zero, she'll conclude that the limit exists, and equals $17$. And you're both wrong, since the limit doesn't exist at all.
For a more complicated function, it may be harder to see that there's one collection of $x$-values where the function does one thing, and a different collection where the function does something very different. That's why you need epsilons and deltas, so you can look at all values of $x$, in a systematic way.