Limits of real functions

## Definition of limits of real functions
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            ## Limit estimation
            Let $f:\mathbb R \to\mathbb R$ and $a\in\mathbb R$.

Informally, $\displaystyle \ell=\lim_{x\to a}f(x)$ is "the value $f(x)$ gets close to when $x$ gets close to $a$".

We need a more formal definition of limits for this course, because we need to be able to *prove* what $\ell$ is.

We need to prove what $\ell$ is because estimating can result in mistakes.

[Excel file demonstrating the dangers of estimating limits](https://nc.dasmithmaths.com/index.php/s/H2HgtQMrWDxbiJB)
 
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 ## Limits of real functions
 #### Definition
 Suppose $c\lt a\lt d$ and $f:(c,a)\cup(a,d):\to\mathbb R$.
 Then

$f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

for all $\varepsilon>0$,   $\exists \delta>0$ such that   $0 \lt \lvert x-a \rvert \lt \delta \implies$ $\lvert f(x)-\ell \rvert < \varepsilon$.

#### Or, in words,
 No matter how small we pick $\varepsilon>0$,

we can always go close enough to $a$ (ie. $\delta$ close) to ensure that

all $x$ closer to $a$ (ie. within $\delta$ of $a$) give
 $f(x)$ within $\varepsilon$ of $\ell$.

#### Notation
            In this case, we write $\displaystyle \lim\_{x\to a} f(x) = \ell$.

[Excel file demonstrating how a limit is defined](https://nc.dasmithmaths.com/index.php/s/b29754YarxgnEsC)