Question

Take the limit of the following question. If the limit doesn't exist state clearly and classify as positive infinity or negative infinity.

Answer 1

`\displaystyle\lim_(x\to9)(\sqrt x-3)/(x-9)=\lim_(x\to9)(\sqrt x-3)/((\sqrt x-3)(\sqrt x+3))=\lim_(x\to9)\frac1{\sqrt x+3}=\frac1{\sqrt9+3}=\frac16`

Alternatively, you can observe that if
`f(x)=\sqrt x`, then by the definition of the derivative, you have


`f'(9)=\displaystyle\lim_(x\to9)(f(x)-f(9))/(x-9)`

You have


`f(x)=\sqrt x\implies f'(x)=\frac1{2\sqrtx}\implies f'(9)=\frac1{2\sqrt9}=\frac16`