11) Given,
[tex]lim_{x\rightarrow3}\frac{x-9}{\sqrt{x}-3}[/tex]
To find: Evaluate.
Solution:
Multiply both numerator and denominator by the conjugate of
[tex]\sqrt{x}+3[/tex]
Now,
[tex]\begin{gathered} lim_{x\rightarrow3}\frac{(x-9)(\sqrt{x}+3)}{(\sqrt{x}-3)(\sqrt{x}+3)} \\ =lim_{x\rightarrow3}\frac{(x-9)(\sqrt{x}+3)}{(\sqrt{x})^2-(3)^2} \\ =lim_{x\rightarrow3}\frac{(x-9)(\sqrt{x}+3)}{x^-9} \\ =lim_{x\rightarrow3}(\sqrt{x}+3) \end{gathered}[/tex]
Put x = 3 in the expression.
[tex](\sqrt{3}+3)=\sqrt{3}(\sqrt{3}+1)[/tex]
Thus, the answer is
[tex]\sqrt{3}(\sqrt{3}+1)[/tex]
