Question

Compute lim x->0 (4x/sin(3x)+x/cos(3x))

Answer 1

\[\lim_{x \rightarrow 0}(\frac{4x}{\sin(3x)}+\frac{x}{\cos(3x)}=\frac{4}{3}\lim_{x \rightarrow 0}\frac{3x}{\sin(3x)}+\lim_{x \rightarrow 0}\frac{x}{\cos(3x)}\]
Recall
\[\lim_{u \rightarrow 0}\frac{u}{\sin(u)}=1\]

Answer 2

the other one is just direct sub.

Answer 3

i didnt quite understand what you just showed =/

Answer 4

Nothing?

Answer 5

okay so what did you do their when you put 4/3

Answer 6

4/3 * 3 =4
I just multiplied the first fraction by 3/3

Answer 7

which became 12x/sin(9x)?

Answer 8

oh no

Answer 9

3sin(3x) does not equal sin(9x)

Answer 10

oh ur right lol sorry about that

Answer 11

whats the purpose of multiplying by 3/3?

Answer 12

\[\lim_{u \rightarrow 0}\frac{u}{\sin(u)}=1\]
Do you recall this?

Answer 13

i dont

Answer 14

so this means we can say
\[\lim_{x \rightarrow 0}\frac{3x}{\sin(3x)}=1\]

Answer 15

alright

Answer 16

so I wanted a 3 up there in our problem

Answer 17

but if i put a 3 on top I have to put a 3 on bottom

Answer 18

yes

Answer 19

so that is why I multiplied by 3/3

Answer 20

i get it so the first part became 1

Answer 21

yes 4/3 times 1 is the first part

Answer 22

not the second part we just plug in 0

Answer 23

now*

Answer 24

is everything ok? you got it?

Answer 25

im writing it down

Answer 26

ok

Answer 27

so then it would be 0/cos(3(0))

Answer 28

0/1=0

Answer 29

yes

Answer 30

so we have 4/3 from the first part+ 0 from the second part
=4/3

Answer 31

oh got it thanks!

Answer 32

:)

Answer 33

Spikey I appreciate that you want to learn this stuff. Thanks for engaging in the learning process. That is really cool dude.

Answer 34

lol thank you :) i take AP calc

Answer 35

in highschool

Answer 36

Go you! :)

Answer 37

It is really cool to have that attitude. You will go far.

Answer 38

theres just some stuff that i dont really comprehend my teacher doesnt teach some stuff clearly

Answer 39

lol yeah i never give up. :)

Answer 40

always trying to learn new things.

Answer 41

Good luck to you and peace out!

Answer 42

lol bye