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# limit question, confusing

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the question is

f(x)=(2x+sin(x))/(2x)

as x approaches 0

split the equation and I get

f(x)=(2x)/(2x) + sin(x)/(2x)

(2x)/(2x)=1?

sin(x)/(2x)=(1/2)? (since sin(x)/x=1)

add the 2 together and I get the answer 3/2

now this is where the trouble starts, when I graph this function, as x approaches 0 from possible and negative side. the y value is 1.00873

so which answer is correct? the 3/2 from algebra or 1.00873 from graphing?

I probally did something wrong somewhere, help me

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Your algebra is correct, even according to my plot.

Perhaps you made a wrong graph?

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I checked, I entered the equation correctly. Graphed it on my Ti-89 with pretty print on so it was easy to check if I entered correctly. this is puzzling

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My plot with both Maple and Graphmatica show clearly that f(0) = 3/2.

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damn it, I found the problem, I was in degree mode. guess any equation with trig fuction in it has to be graphed on radian mode. thanks

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I was about to suggest the very same thing, but obviously you got there before me

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at the risk of sounding INCREDIBLY stupid...

since when was sin(x)/x=1?

Because then you mulitply both sides by x, and then sin(x)=x!!!

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now this is where the trouble starts, when I graph this function, as x approaches 0 from possible and negative side. the y value is 1.00873

so which answer is correct? the 3/2 from algebra or 1.00873 from graphing?

I probally did something wrong somewhere, help me

I think and I hope I know what went wrong.

You didn't turn the angle into radian, right?

at the risk of sounding INCREDIBLY stupid...

since when was sin(x)/x=1?

Because then you mulitply both sides by x, and then sin(x)=x!!!

The case is dealing with limit./

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ok, so explain this to me... when manipulating a formula for a limit, you can say sin(x)/x=1? becase is you take the limit with sin(x)/x, you get sin(0)/0. sin(0)=0, so it's 0/0. But when did we learn that you use different rules when manipulating the formula for taking a limit?

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It's a valid question, BobbyJoeCool and important to follow the solution to this fairly small problem. You solve lim x -> 0 sin(x) /x type of question using L'Hopital's rule (sp?). In other words,

limit of f(x)/g(x) = limit of f'(x)/g'(x)

which is really useful for cases when you get indeterminates like 0/0 or inf/inf using normal rules. So

lim x -> 0 sin(x)/x = sin(0)/0 = 0/0 = indeterminate, so we differentiate the top and bottom and get

lim x -> 0 cos(x)/1 = cos(0)/1 = 1/1 = 1. And therefore, by L'Hopital's rule,

lim x -> 0 sin(x)/x = 1.

Remember your calculus!

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I'm in calc 1... have we haven't "learned" that rule yet (although I know it because my teacher decided to give us a problem I have posted in another thread somewhere). But that makes sence now... thanks!

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I am currently studying some simple limit and function.

If I didn't browse this web, I didn't know I've entered the Calculus branch.

The education system does not fascinate us by the beauty of mathematics.

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in my calculus class, we learned lim x->0 of sin(x) / x is 1 and that lim x -> 0 of cos(x) + 1 / x is 0, however we don't tackle l'hoptal (sp)'s rule until second semester.

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It's a valid question' date=' BobbyJoeCool and important to follow the solution to this fairly small problem. You solve lim x -> 0 sin(x) /x type of question using L'Hopital's rule (sp?). In other words,

limit of f(x)/g(x) = limit of f'(x)/g'(x)

which is really useful for cases when you get indeterminates like 0/0 or inf/inf using normal rules. So

lim x -> 0 sin(x)/x = sin(0)/0 = 0/0 = indeterminate, so we differentiate the top and bottom and get

lim x -> 0 cos(x)/1 = cos(0)/1 = 1/1 = 1. And therefore, by L'Hopital's rule,

lim x -> 0 sin(x)/x = 1.

Remember your calculus! [/quote']

can anyone give the steps to find the derivative of sin(x)

(sin(x+h)-sin(x))/h

what then? use sum double angle formula?

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the derivitive of sin(x)=cos(x)...

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Yes, but he was more interested in how you actually show that using limits

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oh... right. Limit derivaitve. that would be moderatly difficult...

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can anyone give the steps to find the derivative of sin(x)

(sin(x+h)-sin(x))/h

what then? use sum double angle formula?

I tried to figure this out myself just today, couldnt think of how to do it so i just plotted / guess and checked and its cos(x)

someone else would have to explain why its cos(x)

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sorry, double post... the one below is correct (pressed wrong button). If a mod would delete this, that would be nice of them!

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I tried to figure this out myself just today' date=' couldnt think of how to do it so i just plotted / guess and checked and its cos(x)

someone else would have to explain why its cos(x)[/quote']

from the site quoted above by TD...

$\lim_{h \to 0} \frac{sin(x+h)-sin(x)}{h}$

double angle formula

$\lim_{h \to 0} \frac{sin(x)cos(h)+sin(h)cos(x)-sin(x)}{h}$

spilit the limits

$\lim_{h \to 0} \frac{sin(x)cos(h)-sin(x)}{h}+\lim_{h \to 0} \frac{sin(h)cos(x)}{h}$

take variables out (limit will still be the same since when h->0, x might as well be a constant)

$sin(x) \lim_{h \to 0} \frac{cos(h)-1}{h}+cos(x) \lim_{h \to 0} \frac{sin(h)}{h}$

multiply numerator and demoninator of the first one by (cos(x)+1) so you can get cos^2-1 in the numerator which can later be manipulated becaus sin^2+cos^2=1

$sin(x) \lim_{h \to 0} \frac{cos^2(h)-1}{h \cdot (cos(h)+1)}+cos(x) \lim_{h \to 0} \frac{sin(h)}{h}$

manipulates, explained above.

$sin(x) \lim_{h \to 0} \frac{-sin^2(h)}{h \cdot (cos(h)+1)}+cos(x) \lim_{h \to 0} \frac{sin(h)}{h}$

uses a proporty of limits to seporate a bunch of things so you end up with a bunch of do-able limits... (lim a*b= lima * lim b)

$sin(x) \left( \lim_{h \to 0} \frac{-sin(h)}{cos(h)+1} \right) \cdot \left( \lim_{h \to 0} \frac{sin(h)}{h} \right) \cdot \left( \lim_{h \to 0} \frac{1}{cos(h)+1} \right) + cos(x) \lim_{h \to 0} \frac{sin(h)}{h}$

$sin(x) \cdot 0 \cdot 1 \cdot \tfrac{1}{2} + cos(x) \cdot 1=cos(x)$

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in the last step, how did you get $( \lim_{h \to 0} \frac{1}{cos(h)+1})$ ? I can see how -sin^2 split into sin(h) and -sin(h).

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That factor shouldn't be there I believe.

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It shouldn't... I was both copying and doing it myself...it really should be

$sin(x) \left( \lim_{h \to 0} -sin(h) \right) \cdot \left( \lim_{h \to 0} \frac{sin(h)}{h} \right) \cdot \left( \lim_{h \to 0} \frac{1}{cos(h)+1} \right) + cos(x) \lim_{h \to 0} \frac{sin(h)}{h}$

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