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Let ${x_n}$ define a sequence with the following limit:
$$
lim_{n to infty} x_n = a
$$
Let $AA_n$ be a set of straight lines on $XY$ plane each defined by two points $A = (a, a^2)$ and $A_n =(x_n, x_n^2)$. Define a sequence $a_n$ which is obtained by taking the value of abscissa of intersection point of $AA_n$ and $Ox$.

Find:
$$
lim_{nto infty}a_n
$$

I've started from trying to graph what the above means from a geometric perspective. Using Desmos I've crafted a visualization of the sequence and the lines defined in the question section.

From the graph it's crystal clear that the sequence tends to $aover 2$ which gives:

$$
lim_{n to infty} a_n = {aover 2}
$$

The problem is that a graph is not a formal proof. How can I formally find the limit of $a_n$ without involving any graphs?

Easy to see the line $AA_n$ is given by
$$
frac {y_n - a^2}{x_n - a} = frac {x_n^2 - a^2} {x_n - a} = x_n + a
$$
when $x_n neq a$ for all $n$. Then setting $y =0$ in the equation above, we solve for $x$, which is just $a_n$, and obtain
$$
boxed {a_n = a - dfrac {a^2}{x_n + a}}
$$
whenever $x_n neq -a$. Under this condition we let $n to +infty$ and get the result.

If $x_n = pm a$ for some $n$, then such $a_n$ actually is not defined. So we assume $|x_n| neq a$ for all $n$ in the reasoning above.