My inquiry is whether \$x/x\$ is always equal to \$1\$. Ns am greatly intersted in genuine numbers and specifically wonder whether \$x/x\$ is defined at \$x=0\$.

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On one hand the division should leveling to \$1\$, on the other hand you must not be enabled to divide by zero.

I have actually been make the efforts to find whether the simplification "goes first" or even if it is the department causes problem first, yet it has actually proven difficult for me to find usefull search terms.

Note that this question arose after analysis this answer and its an initial comment.  The function \$f(x) = fracxx\$ is characterized for every \$x in historicsweetsballroom.combbRsetminus\$. Its limits exist native the left and also from the ideal at \$0\$, however it is not identified at \$0\$. That doesn"t matter if friend "simplify" very first and then "check" or the other way around, because as you discussed you"re not permitted to divide by \$0\$. Therefore \$\$f(x) = egincases 1 & x eq 0 \ extundefined & x = 0endcases.\$\$Consider the very same question,\$\$g(x) = fracx^2 - 2x + 1x-1.\$\$What is this function?  For any type of real number \$x eq0\$, \$\$x=x,\$\$\$\$Leftrightarrow\$\$\$\$dfracxx=1.\$\$\$x=0\$ \$Rightarrow x=x,\$ yet now we cannot division by \$x\$. (Why?)

In fact, this is one indeterminate form.

Suppose that, \$dfrac00=colorred1\$. Hence, \$0=colorred1cdot0\$.

Therefore, \$colorred2cdot0=colorred1cdot0\$ since \$colorred2cdot0=0\$.

So, \$colorreddfrac21=dfrac00\$.

Thus \$colorred2=1.\$ Note: I"m suspect \$xin R\$ in this totality post.

Indeed \$x/x\$ is only characterized when \$x e 0\$. And also wherever it is defined, its value is \$1\$.

However when people (especially in areas other than historicsweetsballroom.com where historicsweetsballroom.comematics is used) talk about such expressions, frequently what castle really typical is: "The constant (or, much more generally, "well-behaved") duty determined by the values of the given expression where that expression is defined". And also there exists indeed precisely one consistent function \$f:historicsweetsballroom.combb R ohistoricsweetsballroom.combb R\$ so that \$f(x)=x/x\$ because that \$x e 0\$, and that is the consistent function \$f(x)=1\$ for every \$xinhistoricsweetsballroom.combb R\$.

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As a concrete example, consider the function \$sin(x)/x\$ which appears in physics in the amplitude of a wave as soon as describing diffraction top top a slit. This expression is clearly not identified at \$x=0\$. Yet is is of food silly to assume the the amplitude the a wave is not characterized at part point. What physicists actually average is the constant function\$\$f(x)=egincasesfracsin x x & x e 0\1 & x=0endcases\$\$But typically that duty isn"t composed that way; the interpretation is implicitly assumed.