How do I prove what the limit of a function is using the definition of a limit?

Asked Nov 23 '16 at 01:19

Active Nov 23 '16 at 01:21

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Let's say I had to prove the $\lim\limits_{x\to 1}(x^2-3x+4) = 2$. I know that I have to show that for any $k \gt 0$, there exists an $f(k) \gt 0$ such that if $0 \lt |x-2| \lt f(k)$, $|(x^2-3x+4)-2| \lt k$.

I already to $|(x^2 - 3x + 4) - 2|$ to $|x-1|\cdot|x-2|$, but where do I go from here? Thanks in advance!

edited Nov 23 '16 at 01:21

asked Nov 23 '16 at 01:19

clockadd

See the answers to this question or this pdf. – Nov 23 '16 at 01:23
$|x-1|\leq(|x-2|+1)$, so $|x-1||x-2)\leq(f(k)+1)f(k)$. – Luiz Cordeiro Nov 23 '16 at 01:24
Does this answer your question? Does there exist an epsilon-delta proof of the limit for all polynomials? – Oct 09 '23 at 01:02

How do I prove what the limit of a function is using the definition of a limit?

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