What is the common proof for $\lim_{x\rightarrow c}x^2=c^2$.
I tried like this
Let $\varepsilon>0$, want to find a $\delta>0$ such that $\forall x\in\mathbb{R},0<|x-c|<\delta\Rightarrow |x^2-c^2|<\varepsilon$.
Therefore $$-\varepsilon<x^2-c^2<\varepsilon$$
$$c^2-\varepsilon<x^2<\varepsilon+c^2$$
$$-c-\sqrt{c^2+\varepsilon}<x-c<-c+\sqrt{c^2+\varepsilon}$$
Choose $\delta=-c+\sqrt{c^2+\varepsilon}$which proves the problem
Is this proof is correct ? If there is any mistake in it give a correct proof please .thanks
Best Answer
You have made one common arithmetic mistake and two common conceptual mistakes.
The arithmetic and first conceptual mistake are mistakes of algebra. The second conceptual mistake is the first mistake that's actually related to calculus.
The arithmetic mistake is the usual one regarding square roots and forgetting that negative numbers exist — the detail here is a bit complicated; the easiest method would be to go back to your precalculus text and review solving inequalities.
Henceforth, I will assume all the variables are such that your work is correct.
The first conceptual mistake is the usual mistake of proving the converse of what you are trying to solve. You have proven
but what you are trying to prove is
This is a common mistake, because you're taught to solve problems backwards; working backwards is an effective means of simplifying problems, but once you have what you believe to be a solution, you have to turn around and prove that it is so. Again, you can see examples of this in your precalculus class, where you solve equations like $\sqrt{x+2} = x$ but get "spurious" solutions; in this case, $x=-1$.
The other conceptual mistake is that you are trying hard to solve the problem exactly. One of the great things that makes calculus "simple" is that we don't have to solve intervals like this exactly — we can usually make all sorts of simplifications to make our life easier.
The usual trick here is to observe that
$$ \lvert x^2 - c^2\rvert = \lvert x-c\rvert \lvert x+c\rvert $$
so if $\lvert x-c\rvert < \delta$, then
$$ \lvert x^2 - c^2 \rvert = \lvert x-c\rvert \lvert x+c\rvert < \delta \lvert x+c\rvert $$
To deal with the $\lvert x+c\rvert$ term, we simply observe that $x$ is near $c$, so $\lvert x+c\rvert$ shouldn't be much bigger than $\lvert 2c\rvert$.
So, we add a second goal: in addition to wanting to prove
$$ \lvert x - c\rvert < \delta \implies \lvert x^2 - c^2\rvert < \epsilon $$
we also want to have
$$ \lvert x - c\rvert < \delta \implies \lvert x + c\rvert < \lvert 2c\rvert + 1 $$
The latter isn't hard to get: it's enough to pick $\delta \leq 1$.
Thus, if we have both $\lvert x - c\rvert < \delta$ and $\delta < 1$, then we have
$$ \lvert x^2 - c^2\rvert = \lvert x-c\rvert \lvert x+c\rvert < \delta (\lvert 2c\rvert + 1) $$
So if we can additionally choose $\delta$ so that $\delta (\lvert 2c\rvert + 1) \leq \epsilon$, then we have what we need!
Choosing $\delta = \min\left( \frac{\epsilon}{\lvert 2c\rvert + 1}, 1 \right)$ works to satisfy both of the conditions we needed for $\delta$.