Question

can someone thoroughly explain this please?

limit as x approaches 81 f(x)=(sqrtx)-9/x-81

Answer 1

wait...\[{\lim_{x \rightarrow 81}}\frac{ \sqrt{x}-9 }{ x-81 }\] or \[{\lim_{x \rightarrow 81}}\frac{ \sqrt{x-9} }{ x-81 }\]?

Answer 2

nvm you mean the first one right?

Answer 3

Yes

Answer 4

Since substituting 81 in for x will result in an undefined result ...0/0

Answer 5

So you have to use another method...Have you learned l'Hopital's Theorem yet?

Answer 6

No.

Answer 7

hmmmm....

Answer 8

I got this wrong on a quiz so is there another way to do it?

Answer 9

Well what did you put on your quiz?

Answer 10

0

Answer 11

And the correct answer was 1/18, right? Or don't you know?

Answer 12

Yes, that answer was correct. But, I don't know how to get that answer.

Answer 13

Ok ... Wow well. Without using l'Hopital's I am at a loss. What course are you in?

Answer 14

AP Calc. is I'Hopital"s simple? Maybe my teacher assumed that everyone knew it already

Answer 15

OK then you can handle it and that might be the case. Here it is in a nutshell.

Answer 16

If a limit results in 0/0 or infinity/infinity or -infinity/-infinity you can use l'Hoptial's

Answer 17

It says:\[\lim_{x \rightarrow c}\frac{ f(x) }{ g(x)}=\lim_{x \rightarrow c}\frac{ f'(x) }{ g'(x) }\]

Answer 18

So this means that if you can't get the answer by subbing in c for x into the expression b/c it results in 0/0, etc. then take the derivative of the numerator and of the denominator and try again.

Answer 19

So for this problem...\[\lim_{x \rightarrow 81}\frac{ \sqrt{x} -9}{ x-81 }=\frac{ \sqrt{81} -9}{ 81-81 }=\frac{ 0 }{ 0 }\] you can use l'Hopital's

Answer 20

That means...

Answer 21

\[\lim_{x \rightarrow 81}\frac{ \sqrt{x}-9 }{ x-81 }=\lim_{x \rightarrow 81}\frac{ \frac{ 1 }{ 2\sqrt{x} } }{ 1 }=\lim_{x \rightarrow 81}\frac{ 1 }{ 2\sqrt{x} } =\frac{ 1 }{ 2\sqrt{81} }=\frac{ 1 }{18 }\]

Answer 22

How is that @Mynameee ?

Answer 23

Where did the 1/2 come from?

Answer 24

The derivative of \[\sqrt{x}\] is

Answer 25

if you havent covered lhopitals rule you can solve it without it by multiply like this :
\[\frac{ \sqrt{x}-9 }{ x-81 }*\frac{ \sqrt{x}+9 }{ \sqrt{x}+9 }\]
then see what you get , some things will cancel out :)

Answer 26

\[\frac{1}{2\sqrt{x}}\]

Answer 27

Oh! That looks familiar @litchlani

Answer 28

Good call @litchlani

but l'Hopital is so much fun to say!

Answer 29

i know lhopital makes things always so easy but if they havent covered it their prof. probably wanted them to do that

Answer 30

Totally agree ... I am just talking about how much fun it is to say l'Hopital.