formal definition of a limit

In order for this to be true:

$$ \lim_{x\to a}f(x) = L $$

(or, in English: the limit of f(x) approaches L as x approaches a.)

This must also be true:

$$ \forall \epsilon > 0, \exists \delta>0 \text{ such that } 0<|x-a|<\delta \implies |f(x)-L| < \epsilon $$

Now: what does that actually mean?

We know that $|x-a|$ is the distance on our horizontal axis between $x$ and where the limit is being evaluated, and that $|f(x)-L|$ is the distance on our vertical axis between $f(x)$ and the value of the limit.

Therefore, rewritten in layman’s terms, the equation above becomes:

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For ANY positive epsilon, there exists a positive δ, such that:

1. No matter the how small we make δ, the distance between $x$ and where the limit is being evaluated ($a$) is always less than delta but greater than zero.
   1. i.e., this distance can grow INFINITELY small, because no matter how small δ is, this distance will always be smaller.
   2. The stipulation that this distance is greater than zero is to make it so that the function doesn’t actually have to be DEFINED at $x=a$; just that it has to be defined really, really close to $x=a$.

Then, in order for the limit to actually exist, the existence of such a δ must imply that the distance between $f(x)$ and $L$ can similarly become infinitely small, as $|f(x)-L|<\epsilon$ for any positive $\epsilon$, no matter how small.

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TL;DR: in order for a limit to exist at $x = a$, you have to be able to get $f(x)$ really close to the limit $L$ (from either direction) by simply having $x$ being really close to $a$ (from either direction)

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The above conditions must be true in order for the limit to exist!

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also, a funny little note: δ must be written in terms of $\epsilon$! not in terms of $x$ or other such things — delta is a function of epsilon

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basic limit proofs

• EXAMPLE - proving that $\lim\limits_{x\to 1}x =1$
• EXAMPLE - proving that $\lim\limits_{x\to 3}(-5x+4) = -11$
• EXAMPLE - proving that $\lim\limits_{x\to a}x^2 =a^2$
• EXAMPLE - proving that if $\lim\limits_{x\to a}f(x) =L$, then $\lim\limits_{x\to a}|f(x)| =|L|$
• EXAMPLE - proving that if $\displaystyle\lim_{x\to a}|f(x)| = 0$, then $\displaystyle\lim_{x\to a}f(x) = 0$
• EXAMPLE - proving that if $\displaystyle\lim_{x\to a}f(x) = 0$ and $g(x)$ is bounded, then $\displaystyle\lim_{x\to a}f(x)g(x) = 0$

formal definition of one sided limits

The formal definition of $\displaystyle\lim_{x\to a^+}f(x) = L$ is: