Evaluating $lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{sin3theta } } $
$$lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{sin3theta } } $$
My impression:
I don't see how I can change the variables in any way to get it in the form where I can simplify it to 1. I also looked up similar problems and they mention L'Hopital's rule to solve this, which I haven't learned yet. Do not give a straight answer. I already have that available to me.
calculus limits
edited Nov 23 at 17:32
Martin Sleziak
44.6k7115270
asked Jan 20 '15 at 4:06
Cherry_Developer
1,93821431
add a comment |
$$lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{sin3theta } } $$
My impression:
I don't see how I can change the variables in any way to get it in the form where I can simplify it to 1. I also looked up similar problems and they mention L'Hopital's rule to solve this, which I haven't learned yet. Do not give a straight answer. I already have that available to me.
calculus limits
edited Nov 23 at 17:32
Martin Sleziak
44.6k7115270
asked Jan 20 '15 at 4:06
Cherry_Developer
1,93821431
Hint: Break it up into the product of two limits
– ASKASK
Jan 20 '15 at 4:07
add a comment |
$$lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{sin3theta } } $$
My impression:
I don't see how I can change the variables in any way to get it in the form where I can simplify it to 1. I also looked up similar problems and they mention L'Hopital's rule to solve this, which I haven't learned yet. Do not give a straight answer. I already have that available to me.
calculus limits
edited Nov 23 at 17:32
Martin Sleziak
44.6k7115270
asked Jan 20 '15 at 4:06
Cherry_Developer
1,93821431
$$lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{sin3theta } } $$
My impression:
I don't see how I can change the variables in any way to get it in the form where I can simplify it to 1. I also looked up similar problems and they mention L'Hopital's rule to solve this, which I haven't learned yet. Do not give a straight answer. I already have that available to me.
calculus limits
calculus limits
edited Nov 23 at 17:32
Martin Sleziak
44.6k7115270
asked Jan 20 '15 at 4:06
Cherry_Developer
1,93821431
edited Nov 23 at 17:32
Martin Sleziak
44.6k7115270
asked Jan 20 '15 at 4:06
Cherry_Developer
1,93821431
edited Nov 23 at 17:32
Martin Sleziak
44.6k7115270
edited Nov 23 at 17:32
Martin Sleziak
44.6k7115270
edited Nov 23 at 17:32
Martin Sleziak
44.6k7115270
44.6k7115270
asked Jan 20 '15 at 4:06
Cherry_Developer
1,93821431
asked Jan 20 '15 at 4:06
Cherry_Developer
1,93821431
asked Jan 20 '15 at 4:06
Cherry_Developer
1,93821431
1,93821431
Hint: Break it up into the product of two limits
– ASKASK
Jan 20 '15 at 4:07
add a comment |
Hint: Break it up into the product of two limits
– ASKASK
Jan 20 '15 at 4:07
Hint: Break it up into the product of two limits
– ASKASK
Jan 20 '15 at 4:07
Hint: Break it up into the product of two limits
– ASKASK
Jan 20 '15 at 4:07
add a comment |
2 Answers
2
active
oldest
votes
Hint:
$$lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ sin3theta } } = lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ sin3theta } cdot frac{theta}{theta}} = lim_{theta rightarrow 0}{frac{sin(7theta)}{theta} } cdot {frac{theta}{sin(3theta)} }$$
And then do each limit independently.
answered Jan 20 '15 at 4:10
ASKASK
5,87431836
I see how to go from here... but wouldn't it become $lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ 1 } } cdot lim _{ theta rightarrow 0 }{ frac { 1 }{ sin3theta } } $ I don't get where you got the $theta$
– Cherry_Developer
Jan 20 '15 at 4:12
Sorry, I'll make an edit
– ASKASK
Jan 20 '15 at 4:13
Basically all I did was multiply the top and bottom by $theta$
– ASKASK
Jan 20 '15 at 4:15
Sorry, still confused... $lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ 1 } } cdot frac { theta }{ theta } =lim _{ theta rightarrow 0 }{ frac { theta sin { 7theta } }{ theta } } $ no? Where did the $theta$ in the numerator disappear to?
– Cherry_Developer
Jan 20 '15 at 4:17
It went to the otehr limit
– ASKASK
Jan 20 '15 at 4:18
|
show 4 more comments
If you're allowed to use Taylor series, note that $sin(ktheta) approx ktheta$ when $theta$ is near zero:
Solution using this technique, now that I see you probably have solved this:
$$lim_{theta to 0} frac{sin(7theta)}{sin(3theta)} = lim_{thetato0}frac{7theta}{3theta} = frac{7}{3}$$
answered Jan 20 '15 at 4:07
apnorton
15.1k33696
add a comment |
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2 Answers
2
active
oldest
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2 Answers
2
active
oldest
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active
oldest
votes
active
oldest
votes
Hint:
$$lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ sin3theta } } = lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ sin3theta } cdot frac{theta}{theta}} = lim_{theta rightarrow 0}{frac{sin(7theta)}{theta} } cdot {frac{theta}{sin(3theta)} }$$
And then do each limit independently.
answered Jan 20 '15 at 4:10
ASKASK
5,87431836
I see how to go from here... but wouldn't it become $lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ 1 } } cdot lim _{ theta rightarrow 0 }{ frac { 1 }{ sin3theta } } $ I don't get where you got the $theta$
– Cherry_Developer
Jan 20 '15 at 4:12
Sorry, I'll make an edit
– ASKASK
Jan 20 '15 at 4:13
Basically all I did was multiply the top and bottom by $theta$
– ASKASK
Jan 20 '15 at 4:15
Sorry, still confused... $lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ 1 } } cdot frac { theta }{ theta } =lim _{ theta rightarrow 0 }{ frac { theta sin { 7theta } }{ theta } } $ no? Where did the $theta$ in the numerator disappear to?
– Cherry_Developer
Jan 20 '15 at 4:17
It went to the otehr limit
– ASKASK
Jan 20 '15 at 4:18
|
show 4 more comments
Hint:
$$lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ sin3theta } } = lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ sin3theta } cdot frac{theta}{theta}} = lim_{theta rightarrow 0}{frac{sin(7theta)}{theta} } cdot {frac{theta}{sin(3theta)} }$$
And then do each limit independently.
answered Jan 20 '15 at 4:10
ASKASK
5,87431836
I see how to go from here... but wouldn't it become $lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ 1 } } cdot lim _{ theta rightarrow 0 }{ frac { 1 }{ sin3theta } } $ I don't get where you got the $theta$
– Cherry_Developer
Jan 20 '15 at 4:12
Sorry, I'll make an edit
– ASKASK
Jan 20 '15 at 4:13
Basically all I did was multiply the top and bottom by $theta$
– ASKASK
Jan 20 '15 at 4:15
Sorry, still confused... $lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ 1 } } cdot frac { theta }{ theta } =lim _{ theta rightarrow 0 }{ frac { theta sin { 7theta } }{ theta } } $ no? Where did the $theta$ in the numerator disappear to?
– Cherry_Developer
Jan 20 '15 at 4:17
It went to the otehr limit
– ASKASK
Jan 20 '15 at 4:18
|
show 4 more comments
Hint:
$$lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ sin3theta } } = lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ sin3theta } cdot frac{theta}{theta}} = lim_{theta rightarrow 0}{frac{sin(7theta)}{theta} } cdot {frac{theta}{sin(3theta)} }$$
And then do each limit independently.
answered Jan 20 '15 at 4:10
ASKASK
5,87431836
Hint:
$$lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ sin3theta } } = lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ sin3theta } cdot frac{theta}{theta}} = lim_{theta rightarrow 0}{frac{sin(7theta)}{theta} } cdot {frac{theta}{sin(3theta)} }$$
And then do each limit independently.
answered Jan 20 '15 at 4:10
ASKASK
5,87431836
edited Jan 20 '15 at 4:15
answered Jan 20 '15 at 4:10
ASKASK
5,87431836
answered Jan 20 '15 at 4:10
ASKASK
5,87431836
answered Jan 20 '15 at 4:10
ASKASK
5,87431836
5,87431836
I see how to go from here... but wouldn't it become $lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ 1 } } cdot lim _{ theta rightarrow 0 }{ frac { 1 }{ sin3theta } } $ I don't get where you got the $theta$
– Cherry_Developer
Jan 20 '15 at 4:12
Sorry, I'll make an edit
– ASKASK
Jan 20 '15 at 4:13
Basically all I did was multiply the top and bottom by $theta$
– ASKASK
Jan 20 '15 at 4:15
Sorry, still confused... $lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ 1 } } cdot frac { theta }{ theta } =lim _{ theta rightarrow 0 }{ frac { theta sin { 7theta } }{ theta } } $ no? Where did the $theta$ in the numerator disappear to?
– Cherry_Developer
Jan 20 '15 at 4:17
It went to the otehr limit
– ASKASK
Jan 20 '15 at 4:18
|
show 4 more comments
I see how to go from here... but wouldn't it become $lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ 1 } } cdot lim _{ theta rightarrow 0 }{ frac { 1 }{ sin3theta } } $ I don't get where you got the $theta$
– Cherry_Developer
Jan 20 '15 at 4:12
Sorry, I'll make an edit
– ASKASK
Jan 20 '15 at 4:13
Basically all I did was multiply the top and bottom by $theta$
– ASKASK
Jan 20 '15 at 4:15
Sorry, still confused... $lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ 1 } } cdot frac { theta }{ theta } =lim _{ theta rightarrow 0 }{ frac { theta sin { 7theta } }{ theta } } $ no? Where did the $theta$ in the numerator disappear to?
– Cherry_Developer
Jan 20 '15 at 4:17
It went to the otehr limit
– ASKASK
Jan 20 '15 at 4:18
I see how to go from here... but wouldn't it become $lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ 1 } } cdot lim _{ theta rightarrow 0 }{ frac { 1 }{ sin3theta } } $ I don't get where you got the $theta$
– Cherry_Developer
Jan 20 '15 at 4:12
I see how to go from here... but wouldn't it become $lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ 1 } } cdot lim _{ theta rightarrow 0 }{ frac { 1 }{ sin3theta } } $ I don't get where you got the $theta$
– Cherry_Developer
Jan 20 '15 at 4:12
Sorry, I'll make an edit
– ASKASK
Jan 20 '15 at 4:13
Sorry, I'll make an edit
– ASKASK
Jan 20 '15 at 4:13
Basically all I did was multiply the top and bottom by $theta$
– ASKASK
Jan 20 '15 at 4:15
Basically all I did was multiply the top and bottom by $theta$
– ASKASK
Jan 20 '15 at 4:15
Sorry, still confused... $lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ 1 } } cdot frac { theta }{ theta } =lim _{ theta rightarrow 0 }{ frac { theta sin { 7theta } }{ theta } } $ no? Where did the $theta$ in the numerator disappear to?
– Cherry_Developer
Jan 20 '15 at 4:17
Sorry, still confused... $lim _{ theta rightarrow 0 }{ frac { sin { 7theta } }{ 1 } } cdot frac { theta }{ theta } =lim _{ theta rightarrow 0 }{ frac { theta sin { 7theta } }{ theta } } $ no? Where did the $theta$ in the numerator disappear to?
– Cherry_Developer
Jan 20 '15 at 4:17
It went to the otehr limit
– ASKASK
Jan 20 '15 at 4:18
It went to the otehr limit
– ASKASK
Jan 20 '15 at 4:18
|
show 4 more comments
If you're allowed to use Taylor series, note that $sin(ktheta) approx ktheta$ when $theta$ is near zero:
Solution using this technique, now that I see you probably have solved this:
$$lim_{theta to 0} frac{sin(7theta)}{sin(3theta)} = lim_{thetato0}frac{7theta}{3theta} = frac{7}{3}$$
answered Jan 20 '15 at 4:07
apnorton
15.1k33696
add a comment |
If you're allowed to use Taylor series, note that $sin(ktheta) approx ktheta$ when $theta$ is near zero:
Solution using this technique, now that I see you probably have solved this:
$$lim_{theta to 0} frac{sin(7theta)}{sin(3theta)} = lim_{thetato0}frac{7theta}{3theta} = frac{7}{3}$$
answered Jan 20 '15 at 4:07
apnorton
15.1k33696
add a comment |
If you're allowed to use Taylor series, note that $sin(ktheta) approx ktheta$ when $theta$ is near zero:
Solution using this technique, now that I see you probably have solved this:
$$lim_{theta to 0} frac{sin(7theta)}{sin(3theta)} = lim_{thetato0}frac{7theta}{3theta} = frac{7}{3}$$
answered Jan 20 '15 at 4:07
apnorton
15.1k33696
If you're allowed to use Taylor series, note that $sin(ktheta) approx ktheta$ when $theta$ is near zero:
Solution using this technique, now that I see you probably have solved this:
$$lim_{theta to 0} frac{sin(7theta)}{sin(3theta)} = lim_{thetato0}frac{7theta}{3theta} = frac{7}{3}$$
answered Jan 20 '15 at 4:07
apnorton
15.1k33696
edited Jan 20 '15 at 6:12
answered Jan 20 '15 at 4:07
apnorton
15.1k33696
answered Jan 20 '15 at 4:07
apnorton
15.1k33696
answered Jan 20 '15 at 4:07
apnorton
15.1k33696
15.1k33696
add a comment |
add a comment |
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Hint: Break it up into the product of two limits
– ASKASK
Jan 20 '15 at 4:07