Formally proving $\lim_{x\to 1}(x^2+1)=2$ I know this is a really basic question, but I just can't do this limit even while looking at other examples...

I need to prove that $\lim\limits_{x\to 1}(x^2+1)=2$, using delta and epsilon. 

I just can't 'formally' demonstrate this, please help.
For me its just obvious that $x^2$ with $x\to1$ should be $1$, and I understand that epsilon is the difference between points near $x=1$, and delta is how 'near' those f(x's) is to f(x=1). Yet I don't know how to use both of them to reach a proof that $x^2 + 1 = 2$.
edit: How far I got -
$0<|(x^2 + 1)-2|<\epsilon$ ergo $|x-1|<\frac{\epsilon}{|x+1|}$. Let $ \delta = 1$:
$-1 < \epsilon < 1$ so $1<x+1<3$, then : $\frac{\epsilon}{3}<\frac{\epsilon}{x+1}< \epsilon$
This what I worked to discover, now the proof:
$\forall \epsilon>0 \exists \delta>0$:
$0<|x-1|<\delta$ and $|(x^2+1)-2|<\epsilon$, $\delta = min(1, \epsilon)$
Now what?
 A: Let $\varepsilon$ be an arbitrary positive number. If $\left| {x - 1} \right| \le \delta : = \sqrt {\varepsilon  + 1}  - 1$, then
$$
\left| {(x^2  + 1) - 2} \right| = \left| {x^2  - 1} \right| = \left| {x + 1} \right|\left| {x - 1} \right| \le \left| {x - 1} \right|^2  + 2\left| {x - 1} \right| \le \varepsilon .
$$
A: Let $|x -1| <1$;
Then $0<x<2$, and $1<x+1<3$;
Let $\epsilon$ be given
Choose $\delta =\min (1, \epsilon/3).$
Then
$|x-1|<\delta$ implies
$|x^2-1|=$
$|x-1||x+1|<3|x-1|<3\delta\le \epsilon$.
A: The steps to solving your problem are as follows:


*

*Write down the definition of the statement $\lim_{x\to a} f(x) = L$.

*Replace $a, f$ and $L$ in that definition with the actual values you have in your particular problem (in other words, write down the actual statement you need to prove).

*You should now have some sort of statement of the form $\forall \epsilon > 0\exists \delta > 0 \forall x: A\implies B$, where $A$ and $B$ are statements including $x, \epsilon$ and $\delta$.

*Think about the relation $A$, and how it is connected to $B$.


If you still have problems, write (in an edit to your question) how far along the 4 steps you got and where you are now stuck, and we can help you further.
