Question

limit question. lim (x - > 5) 1/(x-5)^4

Answer 1

\[\lim_{x \rightarrow 5} 1/(1+5)^4\]

Answer 2

woops. the 1 is an x

Answer 3

It would be infinity

Answer 4

oh

Answer 5

why so?

Answer 6

Do you know L'Hopital's rule

Answer 7

nope. is it the only way?

Answer 8

the answer is infinity. just have no clue how to work it out.

Answer 9

Ok what you can do is:

Answer 10

wait i just noticed

Answer 11

u mean it's x-5 in the denominator right? otherwise it wouldn't be infinity

Answer 12

yes, x-5

Answer 13

\[\lim_{x \rightarrow 5} \frac{x}{(x-5)^4}\]\[=\lim_{x \rightarrow 5} \frac{x}{(x-5)^4}\]\[=\lim_{x \rightarrow 5} \frac{x}{(x-5)^2(x-5)^2}\]\[=\lim_{x \rightarrow 5} \frac{x}{(x-5)(x-5)(x-5)(x-5)}\]\[=\lim_{x \rightarrow 5} \frac{x/x}{(x/x-5/x)(x/x-5/x)(x/x-5/x)(x/x-5/x)}\]\[=\lim_{x \rightarrow 5} \frac{1}{(1-5/x)(1-5/x)(1-5/x)(1-5)}\]

Answer 14

oops the last factor should also be (1-5/x)

Answer 15

awesome, thanks for your effort! just wondering line 6. why are you diving everything by x?

Answer 16

I hope you see that the bottom factors are (1-1) when substituting x=5, so you get like "1/0" -> infinity

Answer 17

hm it's true that was a useless step

Answer 18

I was thinking that it was going to x->infinity at first, but i realized it was x->5 after lol

Answer 19

ahhh, yes. i see that. thank you. is that a legal move though? because you are changing the function?

Answer 20

for example, (x-5) doesn't equal (1-5/x)

Answer 21

sorry, i guess it is. just would have never thought of that!

Answer 22

Nope :) If you divide the top and bottom by x, it is perfectly valid (it's like you are dividing by "1") Example: \[\frac{2}{3} = \frac{\frac{2}{x}}{\frac{3}{x}}=\frac{2}{x}*\frac{x}{3}=\frac{2}{3}\]

Answer 23

I divided the numerator by x as well: I did x/x = 1 in the numerator, and divided the whole denominator by x

Answer 24

if you divide the numerator, you MUST also divide the denominator!

Answer 25

awesome, this is just amazing. never would have figured out this.

Answer 26

you will learn a neat theorem called "L'hopital's rule" later which will make the computation even easier :)

Answer 27

k. thanks for your time. im now 'a fan' haha.

Answer 28

Hehe awesome ;) Good luck with everything

Answer 29

cheers