Question

Evaluate lim x->pi/6 (sqrt3 cosx)

Answer 1

\[(\sqrt{3} \cos x)\]

Answer 2

Does cos(x) also shares room with 3 in that square root??

Answer 3

Okay.. I got it..

Answer 4

Oh, wait! You only substitute?

Answer 5

No, it doesnt

Answer 6

I am looking for what is a bit tricky part here..

Answer 7

lol

Answer 8

There isnt any eh?

Answer 9

Just put x = pi/6 there.. and you are done..

Answer 10

Till you are getting a finite value on substituting, you can substitute the values directly..

Answer 11

Yeah, i just got that xD

Answer 12

If cos(x) is under root, then also, you will substitute directly..

Answer 13

Yes, exactly :)

Answer 14

Good.. :)

Answer 15

Should I give you one??

Answer 16

That would be great!

Answer 17

Okay: if you want to solve it here, then you can solve it otherwise you can make it your question too and post it also, I leave it on you.. :)

Answer 18

Try:

\[\lim_{x \to -2} \frac{(x^2 - 4)}{(x+ 2)}\]

Answer 19

x is tending to -2 there, be careful there..

Answer 20

-4 :D

Answer 21

Oh..!! I am not good at framing questions... :P

Answer 22

let me give one

Answer 23

@hartnn you wanna try??

Answer 24

limit x->0
of

(sin x+x)/ tan x

Answer 25

No, you're good, but I'm smart :p

Answer 26

Hardik, keep it somewhat simpler one, as I have seen she or he is doing basics of limits.. :)

Answer 27

it is simple fr smart people ;)

Answer 28

and she also know the sin x/x formula!

Answer 29

And she also knows about tan(x)/x, I just told her.. :P

Answer 30

:P

Answer 31

@aroub just take your time, if you have doubt anywhere and then let us know, you don't worry about anything if you are not getting.

But I think, just a small use of your brain will take you there..

Answer 32

Now that's a tricky one!

Answer 33

is it 1?

Answer 34

nopes, try again,

Answer 35

how you got one ? will help u spot the error

Answer 36

Haha, okay!

Answer 37

whenever you see sin x
you should think, how can i use the formula
lim x->0 sin x/x =1

Answer 38

or tan x

because tan x is sin x/cos x and cos 0 =1

Answer 39

Thats what i'm doing

Answer 40

how would u get sin x/x and tan x/x ?

Answer 41

just the "+x" is annoying

Answer 42

you have sin x in the numerator, how would u get the sin x/x ?

Answer 43

okay, wait, let me do this again and i'll come back to you :)

Answer 44

sure :)

Answer 45

Do you get to something like this:

\[\lim x \rightarrow0 \frac{ sinx+x }{ sinx }\]

Answer 46

just replaced tan x by sin x ?
the limiting value will be same as cos 0=1

Answer 47

No, its actually like this:

\[\lim_{x \rightarrow 0} \frac{ sinx+x }{ sinx } \lim_{x \rightarrow 0} cosx\]

Answer 48

but yeah, you're right

Answer 49

yes, your step is correct :)
so now try to find
lim x->0 (sin x+x)/sin x

Answer 50

Can I separate them? like this:
sinx/sinx + x/sinx

Answer 51

sure you can! (there is an easier way, but you can do that too)

Answer 52

now how will you find
lim x->0 x/sin x ?

Answer 53

If i flip (do the reciprocal) everything does the limit change?

Answer 54

please tell me its 2? :p

Answer 55

ofcourse it is 2 !
sorry for the suspense :P

Answer 56

if lim f(x) = a
then lim 1/f(x) = 1/ lim f(x) = 1/a

Answer 57

Hahah, finally! Thank youuuuuu :D

Answer 58

most welcome ^_^
now the easier way

Answer 59

(sin x +x)/ tan x = (sin x/x + x/x)/ (tan x/x)

this i got by dividing 'x' in the numerator and denominator

that i did because i had sin x and wanted sin x/x

Answer 60

after that step, its a piece of cake

sin x/x = 1, x/x = 1
tanx /x =1

Answer 61

That's very funny actually compared to what I did xD

Answer 62

but i am glad you tried it on your own, now you know 2 methods to solve that :)

Answer 63

It can become very simple if you have separated it when you first saw it.. :)

\[\lim_{x \to 0} (\frac{\sin(x)}{\tan(x)} + \frac{x}{\tan(x)}) \implies \lim_{x \to 0}(\cos(x) + \frac{x}{\tan(x)}) \\ \lim_{x \to 0} (\cos(x)) + \lim_{x \to 0}(\frac{x}{\tan(x)})\]

Answer 64

Now you must know that:

\[\lim_{x \to 0} (\frac{\tan(x)}{x}) = \lim_{x \to 0}(\frac{x}{\tan(x)}) = \color{blue}{1}\]

Answer 65

I would've never thought of that haha :p You can also flip it from the beginning lim x->0 tanx/sinx+x or multiply sinx to both the numerator and the denominator so there's kinda 3 methods on solving this :p

Answer 66

Also : \(cos(0) = 1\)..

Answer 67

there are many methods to solve every math problem :3

Answer 68

4 lol

Answer 69

Yes, that's true!

Answer 70

Actually, it is 1 method only, the thing is you are going to once final while playing in between.. :P

Answer 71

*one.

Answer 72

move onto next question :P don't play with this on, the entire day :P

Answer 73

*one

Answer 74

*one final place.. :P

Answer 75

Haha, yes! I'm off to bed anyway. Thank you both so much!

Answer 76

yeah, I am unable to see myself in open list of questions on my left, because we are wandering in closed section of openstudy..

Answer 77

Good Night.. :)