Question

Given that lim x--->2 of the function tan^2x= infinity find values of delta that correspond to M=1000. PLEASE HELP ME :(

Answer 1

this is the proof for the formula. 0< |x-a| < delta then f(x) > M

Answer 2

\[\lim_{x-> 2} \tan^2{x} = \infty \]
find some delta that makes M = 1000...

Can you tell me what delta and M are?

Answer 3

Im not sure how to solve it. Delta isnt given and M isnt given. M is either infinity or 1000, delta just means the distance from the limit. IM CONFUSED.

Answer 4

What I did was I square root 1000 and take the tangent inverse of the square root 1000

Answer 5

the distance from the limit? what does that mean? the distance of WHAT from the limit?
and what is M? I know I can be searching this up but i think you helping me would help us both ahahaha

Answer 6

deltta is the distance from the x value and epsilon is distance from y value. since x is not a value on the graph, the graph has either a hole or discontinuity at 2.

Answer 7

Im a nooby calculus student so hopefully what im saying makes sense. :(

Answer 8

WAIT WAIT WAIT. I MADE A MISTAKE ITS AS X APPROACHES PI/2. IM SORRY.

Answer 9

yup. tan^2(x) is perfectly defined it's not a hole hehe

Answer 10

How do I do tan squared on my calculator? TI-83

Answer 11

uhh what oops this sent a drawing by mistake:
\[\lim_{x->\pi/2} \tan^2 x = \]

Answer 12

i dunno but tan^2 x is just tan x * tan x so (tan x)^2

Answer 13

do i take tan inverse squared of 1000 and then subtract pi/2 from both sides to get the delta because at that point delta will match the epsilon?

Answer 14

I'm not really sure, I''m still unsure what delta and M are, I'm guessing as you approach that value you're going to have some dx and dy, and that dx is delta, and dy is epsilon
and you want a certain dx that makes M = 1000, but i don't know what M is

Answer 15

Like we can ride this function till we reach a point x in which M = 1000?

Answer 16

Do you own the stewart calculus textbook? and yes the function should theoretically be defined at M=1000.

Answer 17

Yes, I checked slader, but they cheated and used a graphing tool. graphing tools are not allowed for this problem my professor said. You can use a graphing tool for analysis but not for answers.

Answer 18

oh aw, in reality you would just use a graphing tool but i see why ahaha

Answer 19

Professor told me that limits is one of the hardest things in beginner calculus. I REALLY WANT TO GET THIS TO EXCEL IN CALCULUS

Answer 20

Let me give you a theorem. Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then limit as x approaches a of fx equals infinity. means that for every positive number M there is a positive number delta such that if 0<|x-a| < delta then fx > M

Answer 21

OK. so if f(x) at a is undefined, then something...CLOSE to a, is M. I get that
so now we want an f(a) = M, or tan^2 x = 1000 within some error delta, or small difference delta where it's not infinity

Answer 22

YES :D

Answer 23

yeah you're good at explaining ahaha now back to the problem i guess

Answer 24

If you solve it riddle me so that I may solve it too without cheating myself. Thank you. <3

Answer 25

yes definitely, same to you! I wanna excel at calculus too hehe

Answer 26

so when 0<|x-2|< delta, then tan^2x >1000

Answer 27

\[\tan^2 (\pi/2 \pm dx) = 1000\]
find dx, hmmm

Answer 28

oops i mea 0<|x-pi/2|<delta

Answer 29

does your dx mean delta x?

Answer 30

yeah dx is just delta, i thought writing it this way is interchangable

Answer 31

So how can we use 0<|x-pi/2|< delta to find x, you think?

Answer 32

yes. I did this and I got x-pi/2 > (sqrt (tan inverse of 1000) - (pi/2))

Answer 33

my answer however does not match the answer in the back.

Answer 34

Huh. Can you explain the procedure you went through to arrive there at least? Maybe I can "debug" it for you and tell you where you went wrong :P

Answer 35

Okay. so I used the formula 0<|x-pi/2|<dx then tan^2x>1000

Answer 36

Though I used a graphing tool to get an approximation and it seems like at x = pi/2 - 0.0316 we get an M = 1000 ahahah but that's not allowed.

Answer 37

now i want to match tan^2(x) to match |x-pi/2|

Answer 38

so in the function tan^2x >1000 I take tan inverse sqaured on both side of the inequality

Answer 39

so x > square root of tan inverse 1000

Answer 40

this is roughly 1.25~

Answer 41

now i subtract pi over two

Answer 42

gives me negative value which is not allowed. because delta >0

Answer 43

we are supposed to arrive at .031 using the formula or else the function is a sham

Answer 44

yeah. I did |tan^-1(1000) - pi/2| < dx as well and arrived at a value dx > 0.000999999666667 which is weird

Answer 45

but we know x > than tan^-1(1000) not is, so that probably introduced a great deal of error there

Answer 46

OMG I SOLVED ITTTT

Answer 47

oops oops i meant |sqrt(tan^-1(1000) - pi/2| < dx which gives dx = 0.317881195141

Answer 48

tan inverse (sqrt(1000)) - (pi/2)

Answer 49

it gives negative value but it is the anwer nevertheless!!! :DDDDDDDDDDDDDDDDDD

Answer 50

so i deduce that the operation we followed to solver for x is improper.

Answer 51

ALL IN A DAYS WORK MY DEAR WATSON! ^_^

Answer 52

wow oops. you're right
\[\tan^2 x = y \rightarrow \tan x = \sqrt y \rightarrow x = \tan^-1 (\sqrt y)\]

Answer 53

ahahahah

Answer 54

yaaaay you did it!

Answer 55

I LOVE YOU! ^_^

Answer 56

and yeah, there was an absolute operation done there~ so being negative doesn't matter

Answer 57

i love you too! ahahaha

Answer 58

thank you. I gotta go and do more math! :D Have a good evening :)

Answer 59

you too! have fun~