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April 30th, 2018, 05:21 PM   #1
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Thumbs up Can you help me to demonstrate these limits? [It's URGENT!!!!]

Hello! Thank you for bothering to help me with this.... I need you to please help me with these points of my calculus activity.

The subject is called Trigonometric Limits.

The first one [A] has to be demonstrated using the sandwich theorem.
And the second [F] has to be demonstrated as you would normally do.

Please show me step by step how you would do it....
Thank you in advance for your help.
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April 30th, 2018, 05:41 PM   #2
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2nd example on this page

The third example should help with the 2nd limit.
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May 1st, 2018, 08:47 AM   #3
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Thank you very much for the first exercise.... It really helped. But you could give me a hand with the second one, because I think they look a little different from each other. I'd really appreciate your help, buddy....
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May 1st, 2018, 09:09 AM   #4
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$\displaystyle \lim_{x \to 0} \dfrac{\tan(kx)}{x}$

$\displaystyle k \cdot \lim_{x \to 0} \dfrac{\tan(kx)}{kx}$

$\displaystyle k \cdot \lim_{x \to 0} \dfrac{1}{\cos(kx)} \cdot \dfrac{\sin(kx)}{kx}$

let $y = kx$

$x \to 0 \implies y \to 0$ ...

$\displaystyle k \cdot \lim_{y \to 0} \dfrac{1}{\cos(y)} \cdot \dfrac{\sin(y)}{y}$

can you finish?
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May 1st, 2018, 09:59 AM   #5
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Quote:
Originally Posted by skeeter View Post
$\displaystyle \lim_{x \to 0} \dfrac{\tan(kx)}{x}$

$\displaystyle k \cdot \lim_{x \to 0} \dfrac{\tan(kx)}{kx}$

$\displaystyle k \cdot \lim_{x \to 0} \dfrac{1}{\cos(kx)} \cdot \dfrac{\sin(kx)}{kx}$

let $y = kx$

$x \to 0 \implies y \to 0$ ...

$\displaystyle k \cdot \lim_{y \to 0} \dfrac{1}{\cos(y)} \cdot \dfrac{\sin(y)}{y}$

can you finish?
could you finish it so I can confirm that I did it right?
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May 1st, 2018, 10:15 AM   #6
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Quote:
Originally Posted by fedeblaze View Post
could you finish it so I can confirm that I did it right?
show what you tried, please
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May 1st, 2018, 10:31 AM   #7
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Originally Posted by skeeter View Post
show what you tried, please
I mean, all I really did was make a mental calculation and I was thinking about applying the fundamental trigonometric identities... but I think that would be too complicated... Then I'd like to know what you did, please.
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May 1st, 2018, 10:46 AM   #8
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Quote:
Originally Posted by fedeblaze View Post
I mean, all I really did was make a mental calculation and I was thinking about applying the fundamental trigonometric identities... but I think that would be too complicated... Then I'd like to know what you did, please.
dude,

Take the last line of Skeeter's post at 10:09

you know what $\lim \limits_{y \to 0} \dfrac{\sin(y)}{y}$ is

you copied it from that website to show it's $1$

now you have a factor of $k \dfrac {1}{\cos(y)}$ left

just plug in the 0 and read it off, the denominator doesn't blow up.
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May 1st, 2018, 07:22 PM   #9
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Originally Posted by fedeblaze View Post
I mean, all I really did was make a mental calculation and I was thinking about applying the fundamental trigonometric identities
Then perhaps you ought to sit down and work them out on paper. Nobody learns material well just by mental calculation.

-Dan
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May 1st, 2018, 08:25 PM   #10
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Then perhaps you ought to sit down and work them out on paper. Nobody learns material well just by mental calculation.

-Dan
Why bother? its way easier to put in zero effort and just ask for answers here.
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