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8.2. TANGENCY
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8.1.15. Problem. Show that $\mathfrak{O}\circ \mathfrak{O}\subseteq \mathfrak{O}$. That is, if $g\in \mathfrak{O}$ and $f\in \mathfrak{O}$, then $f\circ g\in \mathfrak{O}$. (As usual, the domain of $f$ {\it og} is taken to be $\{x:g(x)\in$ dom $f$
8.2. TANGENCY
The fundamental idea of differential calculus is the local approximation of a ``smooth'' function by a translate of a linear one. Certainly the expression ``local approximation'' could be taken to mean many different things. One sense of this expression which has stood the test of usefulness over time is (`tangency''. Two functions are said to be tangent at zero if their difference lies in the family $0$. We can of course define tangency of functions at an arbitrary point (see project
below); but for our purposes, (`tangency at $0$'' will suffice. All the facts we need to know concerning this relation turn out to be trivial consequences of the results we have just proved.
8.2.1. Definition. Two functions $f$ and $g$ in $\mathcal{F}_{0}$ are tangent at zero, in which case we write $f\simeq g$, if $f-g\in 0.$
8.2.2. Example. Let $f(x)=x$ and $g(x)=\sin x$. Then $f\simeq g$ since $f(0)=g(0)=0$ and $\displaystyle \lim_{x\rightarrow 0}\frac{x-\sin x}{x}=\lim_{x\rightarrow 0}(1-\frac{\sin x}{x})=0.$
8.2.3. Example. If $f(x)=x^{2}-4x-1$ and $g(x)=(3x^{2}+4x-1)^{-1}$, then $f\simeq g.$
Proof. Exercise. (Solution .)
8.2.4. Proposition. {\it The relation} ``{\it tangency at zero}'' {\it is an equivalence relation on} $\mathcal{F}_{0}.$
Proof. Exercise. (Solution .)
The next result shows that at most one linear function can be tangent at zero to a given function.
8.2.5. Proposition. {\it Let} $S, T\in L$ {\it and} $f\in \mathcal{F}_{0}$. {\it If} $S\simeq f$ {\it and} $T\simeq f$, {\it then} $S=T.$
Proof. Exercise. (Solution .)
8.2.6. Proposition. {\it If} $f\simeq g$ {\it and} $j\simeq k$, {\it then} $f+j\simeq g+k$, {\it and furthermore}, $\alpha f\simeq\alpha gf0or$ {\it all} $\alpha\in \mathbb{R}.$
Proof. Problem.
Suppose that $f$ and $g$ are tangent at zero. Under what circumstances are {\it hof} and {\it hog} tangent at zero? And when are $f$ {\it oj} and {\it goj} tangent at zero? We prove next that sufficient conditions are: $h$ is linear and $j$ belongs to O.
8.2.7. Proposition. {\it Let} $f, g\in \mathcal{F}_{0}$ {\it and} $T\in L$. {\it If} $f\simeq g$, {\it then} $T\circ f\simeq T\circ g.$
Proof. Problem.
8.2.8. Proposition. {\it Let} $h\in \mathfrak{O}$ {\it and} $f, g\in \mathcal{F}_{0}$. {\it If} $f\simeq g$, {\it then} $f\mathrm{o}h\simeq g\mathrm{o}h.$
Proof. Problem.
8.2.9. Example. Let $f(x)=3x^{2}-2x+3$ and $g(x)=\sqrt{-20x+25}-2$ for $x\leq 1$. Then $f\simeq g.$
Proof. Problem.
8.2.10. Problem. Let $f(x)=x^{3}-6x^{2}+7x$. Find a linear function $T:\mathbb{R}\rightarrow \mathbb{R}$ which is tangent to $f$ at $0.$
8.2.11. Problem. Let $f(x)=|x|$. Show that there is no linear function $T:\mathbb{R}\rightarrow \mathbb{R}$ which is tangent to $f$ at $0.$
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