Question

Find the values of a and b so that the following is true.

Answer 1

I am thinking about L'Hospital's rule but I am not sure how I would apply it...

Answer 2

well, you can separate them,

then solve them

Answer 3

Yeah.... But I have two variables then.

Answer 4

are you sure it tends to infinity?

Answer 5

Yep.

Answer 6

0.

Answer 7

oops.

Answer 8

why two variables?

Answer 9

if you differentiate the variable you get nothing from a, but b will still be present, you can get b right?

Answer 10

\[\lim_{x \rightarrow 0} (\frac{ \sin(2x) }{ x^3 }+a+\frac{ b }{ x^2 })=0\]

Answer 11

Yeah. I think you are right.

Answer 12

ahhh, so it tends to zero?

Answer 13

Yep :P . I typed wrong.

Answer 14

no problem, you can separate them then do the limits,

you can spot it by inspection, very easily

Answer 15

Hmm Let me try...

Answer 16

have you done it?

Answer 17

I can't solve the limit of:

\[\lim_{x \rightarrow 0}\frac{ \sin(2x) }{ x^3 }\]

Answer 18

you can!!

differentiate 3 times until the denominator is 1,

Answer 19

not 1, differentiate until the denominator is a constant

Answer 20

sin(2)=2sin(x)cos(x)

Answer 21

I got -4/3 .

Answer 22

for?

Answer 23

The limit.

Answer 24

sin(2x)/x^3 as x tends to infinity?

Answer 25

how ?

Answer 26

To 0 :P .

Answer 27

lol!!! im losing it today...

Answer 28

but still, i would have thought it was zero

Answer 29

Waitt.

Answer 30

The Limit is infinity..... :( .

Answer 31

yep,

Answer 32

We can't solve anything then.

Answer 33

we could, i mean if one of the variables was infinity

Answer 34

But We can't do anything with infinity.... It's not a number.

Answer 35

wait are you sure it should tend to 0?

Answer 36

Well I would assume it should tend to a finite value.

Answer 37

i suspect the limit should tend to infinity then for it so satisfy the limit

Answer 38

not necessarily,

it can be infinity,

let assume it was infinity, then we can do the question with ease.

have you done analysis by any chance?

Answer 39

No.....

Answer 40

Here is the question.

Answer 41

okay, no problem,
assume it is infinity because i think you were right the first time round,

if it was infinity just plug in infinity to the differential you had computed

Answer 42

If we did that then a would be infinity and b would not exist.

Answer 43

b would, it would be x^2,

Answer 44

one moment,

Answer 45

but since b^2 tends to infinity the whole thing would be 0.

Answer 46

x^2 sorry not b^2.

Answer 47

which would be 0, at when b/x^2 as x ->0

Answer 48

Exactly. We can't work with that.

Answer 49

we could, if b = 0,

Answer 50

0/anynumber = 0

Answer 51

But it's not. B/x^2 is zero, not b.

Answer 52

We don't know what b is.

Answer 53

a could be -infinity,

then this would satisfy the limit

Answer 54

But infinity- infinity is not 0.

Answer 55

yes it is

Answer 56

No it's not. Infinity is not a number. It's a concept.

Answer 57

that is true, i moment, i am trying to multi task,

i jst remembered the proof that it isnt,

Answer 58

http://en.wikibooks.org/wiki/Calculus/Infinite_Limits/Infinity_is_not_a_number

Answer 59

okay, you need to apply l'hpitals rule over and over again, i would recon,

but first you need to put all the equation under one common factor so

\[=\frac{\sin(2x) + ax^{3} + bx}{x^{3}}\]

Answer 60

http://answers.yahoo.com/question/index?qid=20110406112720AARpKn1


here is a better and easier way of doing it

the sint is taylor expansion i think,

the rest is pretty self explanatory

Answer 61

Lol. THanks :P .