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3.1 Difference quotients and limits

Key idea of construction.

Given a function $f:I\rightarrow\mathbb{R}$ and a point $x_{0}\in I$ , a natural question is to ask how $f(x)$ changes when you vary $x$ in a close neighbourhood of $x_{0}$ .

The definition of continuity expresses the idea that the value of $f$ changes little when $x$ changes little.

The definition of a derivative of $f$ expresses the idea how much (or better say, how fast) the value $f(x)$ changes when $x$ changes; it should be described by a rate, something like a difference quotient:

\frac{f(x)-f(x_{0})}{x-x_{0}}.

The closer $x$ gets to $x_{0}$ , the more the value of the difference quotient approaches the actual rate of change in $x_{0}$ , i.e., the slope of the graph of $f$ in $x_{0}$ . Notice that we cannot have $x=x_{0}$ because then the denominator would be $0$ ; we must rather have $x$ very close to $x_{0}$ . What concept does this remind you of?

Let us provide a precise definition of the derivative:

Definition 3.1.1 (Derivative).

Given a function $f:I\rightarrow\mathbb{R}$ on an interval $I$ and $x_{0}\in I$ . We say $f$ is differentiable at $x_{0}$ if

\mbox{there is $m\in\mathbb{R}$ such that }\frac{f(x)-f(x_{0})}{x-x_{0}}% \rightarrow m,\mbox{ as }x\rightarrow x_{0},

and in this case the derivative of $f$ at $x_{0}$ is $f^{\prime}(x_{0}):=m$ , sometimes also denoted by $\displaystyle\Big{(}\frac{\operatorname{d}}{\operatorname{d}x}f\Big{)}(x_{0})$ .
$f$ is said to be differentiable if it is differentiable at every $x_{0}\in I$ . By iteration, we write $f^{\prime\prime}:=(f^{\prime})^{\prime}$ for the second derivative of $f$ (if it exists), and $f^{(n)}$ for the $n$ -th derivative.

Exercise L.

How would you write the above definition of differentiability in terms of $\varepsilon$ and $\delta$ ? Remember: arrows are always a short-hand notation for a precise $\varepsilon$ - $\delta$ criterion.

Example 3.1.2.

(1)

Consider the function

$h:\mathbb{R}\rightarrow\mathbb{R},\quad h(x)=x.$

Then we have

$\frac{h(x)-h(x_{0})}{x-x_{0}}=\frac{x-x_{0}}{x-x_{0}}=1.$

This converges to $1$ as $x\rightarrow x_{0}$ , which proves the differentiability of $h$ in $x_{0}$ , with $h^{\prime}(x_{0})=1$ .
(2)

Consider the function

$f:\mathbb{R}\rightarrow\mathbb{R},\quad f(x)=x^{2}.$

Then at $x_{0}$ , we have

$\frac{f(x)-f(x_{0})}{x-x_{0}}=\frac{x^{2}-x_{0}^{2}}{x-x_{0}}=x+x_{0}$

This converges to $2x_{0}$ as $x\rightarrow x_{0}$ , which proves the differentiability of $f$ in $x_{0}$ and that $f^{\prime}(x_{0})=2x_{0}$ .
(3)

Consider the function

$g:\mathbb{R}\rightarrow\mathbb{R},\quad g(x)=|x|.$

We know from MATH113 that this is continuous. We consider the difference quotient with $x$ sufficiently close to $x_{0}$ because eventually we are interested in the limit $x\rightarrow x_{0}$ . (By “sufficiently close” we mean that e.g. $|x-x_{0}|<|x_{0}|$ if $x_{0}\not=0$ , so that $x$ and $x_{0}$ lie on the same side of $0$ .) Let us distinguish three cases:

Case $x_{0}>0$ :

In this case we may assume that $x>0$ as well (because we said $x$ close to $x_{0}$ ) and we get

$g(x)=\frac{|x|-|x_{0}|}{x-x_{0}}=\frac{x-x_{0}}{x-x_{0}}=1,$

proving differentiability in $x_{0}>0$ .
Case $x_{0}<0$ . In this case we may assume that $x<0$ as well and we get

$g(x)=\frac{|x|-|x_{0}|}{x-x_{0}}=\frac{-x+x_{0}}{x-x_{0}}=-1,$

proving differentiability in $x_{0}<0$ .
Case $x_{0}=0$ . In this case we must consider both $x>0$ and $x<0$ because we only know that $x$ is close $x_{0}=0$ . We have

$g(x)=\frac{|x|-|x_{0}|}{x-x_{0}}=\left\{\begin{array}[]{l@{\; :\;}l}\frac{x}{x% }=1\hfil\;:\;&x>0\\ \frac{-x}{x}=-1\hfil\;:\;&x<0.\end{array}\right.$

Then the left-sided limit of the difference quotient is $-1$ in contrast to the right-sided $+1$ . Thus $g$ is not differentiable at $x_{0}=0$ .
(4)

The functions $\sin,\cos:\mathbb{R}\rightarrow\mathbb{R}$ are differentiable. This is quite cumbersome to prove and we simply assume it henceforth. A rigorous approach will be discussed in MATH210.