Limit Laws

L’Hôpital’s Rule §

$$\underset{x\to c}{\lim }\frac{f(x)}{g(x)}=\underset{x\to c}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}$$

Inapplicable Scenarios

Sure, here are the limit laws in markdown table format:

Laws	Description
Constant Law	${\lim }_{x\to a}c=c$
Identity Law	${\lim }_{x\to a}x=a$
Sum/Difference Law	${\lim }_{x\to a}[f(x)\pm g(x)]={\lim }_{x\to a}f(x)\pm {\lim }_{x\to a}g(x)$
Product Law	${\lim }_{x\to a}[f(x)\cdot g(x)]={\lim }_{x\to a}f(x)\cdot {\lim }_{x\to a}g(x)$
Quotient Law	${\lim }_{x\to a}\frac{f(x)}{g(x)}=\frac{{\lim }_{x\to a}f(x)}{{\lim }_{x\to a}g(x)}$ (if ${\lim }_{x\to a}g(x)\ne 0$)
Exponentiation Law	${\lim }_{x\to a}[f(x){]}^n=[{\lim }_{x\to a}f(x){]}^n$
Composition Law	${\lim }_{x\to a}[f(g(x))]=f({\lim }_{x\to a}g(x))$
Squeeze Theorem	If $f(x)\le g(x)\le h(x)$ for all $x$ in some interval containing $a$, and ${\lim }_{x\to a}f(x)=L={\lim }_{x\to a}h(x)$, then ${\lim }_{x\to a}g(x)=L$.
Intermediate Value Theorem	If $f(x)$ is a continuous function on the closed interval $[a,b]$ and $C$ is any number between $f(a)$ and $f(b)$, then there exists at least one number $c$ in the interval $(a,b)$ such that $f(c)=C$.