I am a university student and we learned about 'Formal Definition of Limit' this week.
Now I have a question about that topic, and it prevents me to understand the concept.
For example, we can prove a limit we know is true that way;
$$\begin{align*} &\lim_{x\to3}2x+1=7\\ &to\:prove\:that;\\ &|f(x)-L|<\epsilon\\ &\Rightarrow|2x+1-7|<\epsilon\\ &\Rightarrow|2x-6|<\epsilon\\ &\Rightarrow2|x-3|<\epsilon\\ &\Rightarrow|x-3|<\frac{\epsilon}{2} \end{align*}$$ So we can assume $\delta=\frac{\epsilon}{2}$. Now we can say there is a $\delta$ for every $\epsilon$. And is always greater than $|x-3|$. And so we can say the limit is true.
But what if we tried to proof a limit which we know it is already false. $$\begin{align*} &\lim_{x\to3}2x+1\not=8\\ &to\:prove\:that;\\ &|f(x)-L|<\epsilon\\ &\Rightarrow|2x+1-8|<\epsilon\\ &\Rightarrow|2x-7|<\epsilon\\ &\Rightarrow|2(x-3)-1|<\epsilon\\ &\Rightarrow-\epsilon<2(x-3)-1<\epsilon\\ &\Rightarrow-\epsilon+1<2(x-3)<\epsilon+1\\ &\Rightarrow\frac{-\epsilon+1}{2}<x-3<\frac{\epsilon+1}{2} \end{align*}$$ In this example I couldn't finish it. How can we proof that this limit is not equal to 8?