$$\def \C {\mathbb{C}}\def \R {\mathbb{R}}\def \N {\mathbb{N}}\def \Z {\mathbb{Z}}\def \Q {\mathbb{Q}}\def\defn#1{{\bf{#1}}}$$
Common mistake
Probably the most common mistake about the definition of limit is to describe it as `the thing that the sequence gets closer to but never reaches’. I have seen this so many times. The first mistake, of course, is that this is not the precise definition given. (Also, using the word ‘thing’ in any definition is a sure sign that there is something wrong!)

The second mistake is that it is not even true. Elements of the sequence can equal the limit and hence the limit is ‘reached’. Not only that, they can equal the limit in number of ways. In each of the following examples the limit is equal to zero.

1. No elements equal the limit. E.g., $a_n=1/n$.
2. A finite number of elements equal the limit. E.g., $a_n=\frac{1}{n}-\frac{3}{n^2}+\frac{2}{n^3}$.
3. An infinite number of elements equal the limit. E.g., $a_n=\frac{\cos (n\pi /2 ) }{n}$.
4. All elements equal the limit. E.g., $a_n=0$.

Another Mistake
A second, though less common, mistake is to think that the sequence steadily approaches a limit from one direction. True, we see many examples where this happens. For example, $1/n$ tends to its limit $0$ by approaching in a manner where each element is closer than the last.

This need not happen.

Consider the example above, $a_n=\frac{\cos (n\pi /2 ) }{n}$. Here the sequence ‘jumps about’. An element may be further from the limit than the one before it. However, as the sequence progresses the elements do get closer and closer.

Notation mistakes
A couple of other mistakes are essentially to do with notation.

1. Error: ‘We have $a_n\to a$ for large $n$’.
   This seems to be a confusion with approximations. E.g., $n^2+2n+3$ is approximately $n^2$ for large $n$.

   True, in the concept of limit we are investigating what happens as $n$ gets large but the correct way to say the above is ‘We have $a_n\to a$ as $n\to \infty $’.

2. Error: ‘$\displaystyle \lim_{n\to\infty} a_n\to a$’. The error here is that the left hand side is a single number not a sequence. We can only say a sequence tends to $a$. We can’t say that a number tends to $a$.

   The correct way is to write: $\displaystyle \lim_{n\to\infty} a_n=a$.