3.1 Section 1.1

3.1.1 definition of limit of function \(f\relax (x) \)

We say that the limit to \(f\relax (x) \) is \(L\) as \(x\) gets close to \(c\), if for each number \(\varepsilon \) we can find another number \(\delta \) such that \(\left \vert f\relax (x) -L\right \vert <\varepsilon \) for all \(x\) within a distance \(\delta \) from \(c.\)

So if we change \(\varepsilon \), may be make it smaller, we need to find another \(\delta \), most likely smaller than before also, such that \(\left \vert f\left ( x\right ) -L\right \vert <\varepsilon \) inside this new interval around \(c\)

Figure 3.1:Limit

note: We say \(\lim _{x\rightarrow c}f\relax (x) \) exist if \(\lim _{x\rightarrow c^{-}}f\relax (x) =\lim _{x\rightarrow c^{+}}f\relax (x) =L\)

Example of a function where \(\lim _{x->0}f\relax (x) \) does not exist is \(f\relax (x) =\left \{ \begin {array} [c]{c}-1\ \ x<0\\ 0\ \ x=0\\ +1\ \ x>0 \end {array} \right . \)

Figure 3.2:example

note: A function can be defined at a point, but not have a limit at that point (as the example above shows)

3.1.2 Definition of continuous function at a point

A function \(f\relax (x) \) is continuous at \(x=c\) if it is defined at that point, and if \(\lim _{x->c}f\relax (x) \) exist and is equal to \(f\relax (c) \)

Example: of a function that has \(\lim _{x->c}f\relax (x) \) exist, but \(f\relax (c) \)  is not equal to this limit. hence not continues at \(x=c\)

Figure 3.3:example

Example of function where \(f\relax (c) \) equal the limit at \(x=c\), and \(\lim _{x->c}f\relax (x) \) exist, hence continues

Figure 3.4:example

3.1.3 Definition of derivative of function \(f\relax (x) \) at \(c\)

if \(f\relax (x) \) is continues at \(x=c\), then \[ f^{\prime }\relax (c) =\lim _{x\rightarrow c}\frac {f\relax (x) -f\relax (c) }{x-c}\]

note: The above \(f^{\prime }\relax (c) \) is defined only if the limit exist and is the same as we approach \(c\) from either side.

Conversely, we say that a function \(f\relax (x) \) is differentiable at \(x=c\) iff \(f^{\prime }\relax (c) \) exist and \(f\relax (c) \) is continues.

In other words, \(f\relax (x) \) is differentiable at \(x=c\) iff \(\lim _{x\rightarrow c^{-}}\frac {f\relax (x) -f\relax (c) }{x-c}=\lim _{x\rightarrow c^{+}}\frac {f\relax (x) -f\relax (c) }{x-c}=f^{\prime }\relax (c) \)

note: It is possible for a function to be continues at \(c\) but not be differentiable there if the above limit is not the same as we approach \(c\) from either side.

Example, \(f\relax (x) =\left \vert x\right \vert \)

Figure 3.5:example

Figure 3.6:example

3.1.4 Intermediate value theorem

on interval \([a,b]\), a continues function assumes all values between \(f\left ( a\right ) \) and \(f\relax (b) \)