Question

L’Hˆopital’s Rule: If $f$and $g$are ____ and $g\left(x\right)$is nonzero near $x=c$, and if $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ is of the indeterminate form ____ or ____, then$\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\_\_\_\_\_$.

Short answer

L’Hˆopital’s Rule: If $f$and $g$are differentiable functions and $g\left(x\right)$is nonzero near $x=c$, and if $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$is of the indeterminate form $\frac{0}{0}$or $\frac{\infty }{\infty }$, then localid="1648554145141" $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to c}{\lim }\frac{f^{\text{'}}\left(x\right)}{g^{\text{'}}\left(x\right)}$.

Step-by-step solution

Step 1. Given information  

The given statement is,

L’Hˆopital’s Rule: If $f$and $g$are ____ and $g\left(x\right)$is nonzero near $x=c$, and if $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$is of the indeterminate form ____ or ____, thenlocalid="1648554437236" $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\_\_\_\_\_\_$.

Step 2. Fill in the blanks  

The given statement is a theorem

That is,

L’Hˆopital’s Rule: If $f$and $g$are differential equation and $g\left(x\right)$is nonzero near $x=c$, and if $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$is of the indeterminate form $\frac{0}{0}$or $\frac{\infty }{\infty }$, then localid="1648554795208" $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to c}{\lim }\frac{f^{\text{'}}\left(x\right)}{g^{\text{'}}\left(x\right)}$.