Give a formula for the extended function that is continuous at the indicated point. f(x)=(x-4)/(sqrt(x)-2), x=4

Question

mathmath333 · Answer

$\Large\rm \lim_{x\to 4}\dfrac{x-4}{\sqrt x-2}=f(4)$ 

$\Large\rm =\lim_{x\to 4}\dfrac{(\sqrt x)^2-2^2}{\sqrt x-2}$

$\Large\rm =\lim_{x\to 4}\dfrac{[(\sqrt x)-2][\sqrt x)+2]}{\sqrt x-2}$  

$\Large\rm =\lim_{x\to 4}\sqrt x+2$
  
$\Large\rm =\sqrt 4+2$  

$\Large\rm =2+2=4$  

$\Large\rm or$

$\Large\rm =-2+2=0$

$\Large\rm f(x)=\begin{cases}
\dfrac{x-4}{\sqrt x-2}, x\ne 4\\
x=4,~~~~ x=4\


x=0,~~~~x=4
\end{cases}$