Question

I am trying to figure out how to properly use algebra to prove that the limit as x->0 of (cos x)/x does not exist. Anyone able to give me some advice?

Answer 1

when you evaluate limits you can plug in (zero in this case) where there is an 'x', so the limit in this case would be cos(0)=1 over 0. and any fraction with a zero in the denominator is 'undefined'....if that helps

Answer 2

That is why I can't use direct substitution. However, that's the beginning of the problem. I can't use the limit properties to factor out 1/x either because that limit also does not exist.

Answer 3

another thing you can use is the formal definition of limits to evaluate that. have you gone over that?

Answer 4

thats the only thing i can think of that will let you use algebra to evaluate the limit

Answer 5

Yes we have gone over the formal definition of limits. I fail to see how that helps, unfortunately. I know what the graph looks like. I know that the one sided limits don't agree and that is why it does not exist. I need to algebraically prove that, though.

Answer 6

hmmmm, let me use the formal definition, see what happens

Answer 7

ok, if you use the limit definition you can show using algebra (and trig identities) that the limit is 1/0 which is undefined

Answer 8

just because something is undefined doesn't mean that there isn't a limit, though. Example: sin x/x as x approaches zero

Answer 9

the math tutors had a hard time with this one at school too sadly

Answer 10

i know, you can use L'Hopitals rule to show that the limit as x->0 of cos(x)/x exists....to show that it doesnt exist using algebra though, the only way i can think of is using the formal definition. im not sure what else to do

Answer 11

I am not allowed to use that yet

Answer 12

L'Hopitals

Answer 13

I can retry but I have been stumped too long to still have hope lol

Answer 14

the only other thing you can try is to find values to the left of 0 and to the right of 0. (one sided limits)

Answer 15

I thought about a table of values.

Answer 16

yea, something like that lim x->-0.001 cosx/x
and lim x-> +0.001 cosx/x

Answer 17

this way the values are close to 0 but never actully reach zero

Answer 18

There is a batter way, though. My teacher is OCD with the ink.

Answer 19

better*

Answer 20

sorry i couldnt have helped better. like i said, the formal definition seems the most algebra way to solve your problem. but i guess its really upto what your teacher wants

Answer 21

I don't think it was solved using the formal definition. Maybe it can be, but you described undefined and that just isn't the way to tell, unfortunately. Thanks, though.

Answer 22

I tried lim from the left and lim from the right to show unbounded behavior and that the one sided limits don't agree. Does this sound legit? Or maybe I am not using algebra properly. Ugh. Not sure.