Does $\lim_{x\to\infty} f(x) = 1$ imply that $f(x) = 1$ when $x = \infty$? If $\lim_{x\to\infty} f(x) = 1$, then is it correct to state it as $f(x) = 1$ when $x = \infty$?
Can we define a function in this way, if not why? Also if a limit exists at a point $x$ then does the function always has to be defined for that  $x$ with the same value obtained through the limit?
 A: Remember, a limit is what happens 'as you approach something' not 'once you get there'. Meaning, the function does not even have to be defined there. For example, $\displaystyle \lim_{x \to 0} x^2= 0$ and if $f(x)=x^2$, we have $f(0)=0$. But for $f(x)= \frac{\sin x}{x}$, we have
$$
\lim_{x \to 0} \dfrac{\sin x}{x}= 1
$$
But $f(0)$ is not defined.
Now you undoubtably have seen if a function is failed to be defined at a value but the limit exists there, we have a removable discontinuity that we can 'fill in' by defining it there. So we can define
$$
g(x)=
\begin{cases}
\dfrac{\sin x}{x}, & x \neq 0 \\
1, & x=0
\end{cases}
$$
Now $g(x)$ is continuous. This seems to be what you're thinking about in your question. However, you are working with limits over the real numbers so $x \in \mathbb{R}$, but $\pm \infty \notin \mathbb{R}$. So it makes no sense for you to define a function at $\pm \infty$, instead merely to talk about what happens to a function $f(x)$ as $x \to \pm \infty$.
A: If, say, $f$ is a function from $(0,\infty)$ to $\Bbb R$ and if $\lim_{x\to\infty}f(x)=1$, then, no you cannot say that $f(\infty)=1$ because $f(x)$ is undefined if $x=\infty$; $f(x)$ is defined only when $x\in(0,\infty)$.
And if $f\colon\Bbb R\longrightarrow\Bbb R$ is defined by$$f(x)=\begin{cases}1&\text{ if }x\ne0\\0&\text{ if }x=0,\end{cases}$$then $\lim_{x\to0}f(x)=1$, but $f(0)=0$.
A: No.  The notation $\lim_{x\to\infty}f(x) = 1$ has a very precise meaning.  It means that given any $\varepsilon>0$ we can always find a real number $c$ such that for all $x > c$ we have $|f(x) - 1| < \varepsilon.$  Note that nowhere in there do we know what the exact value of $f(x)$ is at a particular point $x$.
Beyond that, $\infty$ is not a number, so it doesn't make sense in this context to define $f(\infty)$ or say $x = \infty$.
To answer your last question, no, we don't require functions to be defined so that $\lim_{x\to a}f(x) = f(a)$.  This means that the function is continuous at $x = a$, which is not true of all functions.
