Why are the graphs are different for $\frac{(x-9)}{{\sqrt x}-3}$ and ${\sqrt x}+3$? I am very curious why $\frac{(x-9)}{{\sqrt x}-3}$ and ${\sqrt x}+3$ yields different values (or one is undefined while other gives $6$) at $x=9$ even they are equal.
$$
\frac{x-9}{{\sqrt x}-3} = \frac{{({\sqrt x} + 3)({\sqrt x} - 3)}}{{\sqrt x}-3}
= {\sqrt x}+3
$$
And some graph plotting websites give the same graph for both functions while google search gives different graphs for the two functions. 
 A: Both functions $\dfrac{x-9}{{\sqrt{x}}-3}$ and ${\sqrt{x}}+3 $ agree over $[0,9)\cup (9,\infty)$, but the first one is not defined at $x=9$ whereas the the latter is.
A: Since division by zero is undefined, your assertion that
$$\frac{x - 9}{\sqrt{x} - 3} = \frac{(\sqrt{x} + 3)(\sqrt{x} - 3}{\sqrt{x} - 3} = \sqrt{x} + 3$$
is only true when $x \neq 9$.  The function 
$$f(x) = \sqrt{x} + 3$$
has domain $[0, \infty)$, while the function
$$g(x) = \frac{x - 9}{\sqrt{x} - 3}$$
has domain $[0, 9) \cup (9, \infty)$ since division by zero is undefined.  Since the functions $f$ and $g$ have different domains, they are not the same even though they agree on the intersection of their domains.  The graph of $f$ is continuous, while the graph of $g$ has a hole at the point $(9, 6)$.  Therefore, the graphs of $f$ and $g$ are indeed different.  
A: Apparently your question arises in the case of evaluating the limit:
$$\lim_{x\to 9} \frac {x-9}{\sqrt x - 3}.$$ Evaluating the function at $x=9$ yields an indeterminate form $\frac 00$, which does not mean the limit is undefined; what it tells us is that more work needs to be done, in order to understand the behavior of the function as $x$ becomes arbitrarily close to $9$, acknowledging $x \neq 9$. 
So we factor out and cancel factors in this case, in order to evaluate the limit as $x\t 9,\;x\neq 9$. T
he cancelation is valid for all $x\neq 9$, so we can do this.  
So in this sense, the initial  limit is equivalent to $$\lim_{x\to 9} \sqrt x + 3 = 6$$
This is the graph wolfram alpha provides upon entry of $$\lim_{x\to 9} \frac {x-9}{\sqrt x - 3}:$$

