Question

lim
x->0
of
sin(2x)sin(7x)/xsin(9x)

evaluate the limit.
please help?

Answer 1

combine the x in numerator with either sin2x or sin 7x
then multiply and divide by x

Answer 2

i would go with my first guess of \(\frac{2\times 7}{9}\) but i suppose that is not a good explanation

Answer 3

\[\lim_{x \to 0}\frac{\sin(ax)}{\sin(bx)}=\frac{a}{b}\]

Answer 4

do you have to do each limit individually?

Answer 5

damn typo

Answer 6

\[\lim_{x\to 0}\frac{\sin(2x)}{\sin(9x)}\times \lim_{x\to 0}\frac{\sin(7x)}{x}\]

Answer 7

you can use the product of the limits

Answer 8

why are you multiplying the denominator with the numerator?

Answer 9

hope it is clear that the first one is
\[\frac{2}{9}\] and the second is 7

Answer 10

i am not unless i made another mistake

Answer 11

did you mix up the sin7x with the sin9x?

Answer 12

\[\frac{\sin(2x)\sin(7x)}{x\sin(9x)}\] is what you start with right?

Answer 13

correct : )

Answer 14

you can break it up into any product you choose.
i chose
\[\frac{\sin(2x)}{\sin(9x)}\times \frac{\sin(7x)}{x}\] but you could split it another way if you like

Answer 15

i don't understand why the sin(9x) is now under the sin(2x) instead of sin(7x)

Answer 16

either way right?

Answer 17

\[\frac{ab}{cd}=\frac{a}{c}\times \frac{b}{d}=\frac{a}{d}\times \frac{b}{c}\]

Answer 18

pick one it makes no difference which one you choose to use

Answer 19

so now i multiply the first limit by 2/9 and the second limit by 7/7?

Answer 20

the idea is this:
\[\lim_{x\to 0}\frac{\sin(ax)}{x}=a\] and
\[\lim_{x\to 0}\frac{\sin(bx)}{\sin(cx)}=\frac{b}{c}\]

Answer 21

so you are going to get
\[\frac{2}{9}\times 7\] or if you do it the other way you are going to get
\[2\times \frac{7}{9}\] makes no difference

Answer 22

right, so i should end up with lim x-> 0 of \[\frac{ \sin2x }{ \sin9x }\times \frac{ 2 }{ 9 }+\frac{ \sin7x }{ x }\times \frac{ 7 }{ 7 }\]

Answer 23

right but you need a times sign not a plus sign

Answer 24

bring the 2/9 infront of the limit sign, same with the 7 and end up with 2/9times7?

Answer 25

so it should be 2/9t x 7?

Answer 26

yes, without the \(t\)

Answer 27

answer being 14/9?

Answer 28

yes

Answer 29

thank you very much for your help! : )

Answer 30

yw