Limit

[nycmath]

See attachment. Enjoy.

[Math Tutor]

Factor the numerator as a difference of squares: [tex]x-9=(\sqrt{x}-3)(\sqrt{x}+3)[/tex].

For [tex]x \neq 9[/tex] the factor [tex]\sqrt{x}-3[/tex] cancels:

[tex]\frac{x-9}{\sqrt{x}-3}=\frac{(\sqrt{x}-3)(\sqrt{x}+3)}{\sqrt{x}-3}=\sqrt{x}+3[/tex]

Hence [tex]\lim_{x \to 9}\frac{x-9}{\sqrt{x}-3}=\sqrt{9}+3=6[/tex].

(Same thing: multiply top and bottom by the conjugate [tex]\sqrt{x}+3[/tex].)

[Eigenvalue]

[tex]\lim_{x \to 9}x[sqrt(x)+3]
=[tex]\sqrt{9}[/tex]+3=6

[nycmath]

Factor the numerator as a difference of squares: [tex]x-9=(\sqrt{x}-3)(\sqrt{x}+3)[/tex].

For [tex]x \neq 9[/tex] the factor [tex]\sqrt{x}-3[/tex] cancels:

[tex]\frac{x-9}{\sqrt{x}-3}=\frac{(\sqrt{x}-3)(\sqrt{x}+3)}{\sqrt{x}-3}=\sqrt{x}+3[/tex]

Hence [tex]\lim_{x \to 9}\frac{x-9}{\sqrt{x}-3}=\sqrt{9}+3=6[/tex].

(Same thing: multiply top and bottom by the conjugate [tex]\sqrt{x}+3[/tex].)

Pretty easy and straightforward.

[nycmath]

[tex]\lim_{x \to 9}x[sqrt(x)+3]
=[tex]\sqrt{9}[/tex]+3=6

Correct but please edit your LaTex.