Stuck on a possibly simple trig limit

  • Thread starter rambo5330
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  • #1
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Homework Statement



how do you go about solving the equation
lim (t->0) 2t / (sin(t)) - t

the answer in the text is significantly different than what i get.. i can get most of the other trig limits using the fundamental limit etc.. but this one im stuck ? i may be way over complicating it but i think it needs some algebraic manipulation in the current form?

Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
558
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You should notice that [tex]sin \textit{x} /\textit{x} [/tex] is a notable limit
 
  • #3
9
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lim x->0 of sinx/x =1, so
lim x->0 of 1/(sinx/x) is also 1, and 1/(sinx/x)=x/sinx, so
lim x->0 of x/sinx = 1
from here just rearrange the equation.
 
  • #4
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[STRIKE][STRIKE][/STRIKE][/STRIKE]yes I'm aware of sinx/x = 1 and all the other variations of it etc thats what i was refering to as the fundamental trig limit but i think its the fact its in the form. sinx - x i cannot factor out an x and since its in the form of a binomial this is where i think im just doing something stupid with my algebra any pointers on that one?

rember its 2t / (sin(t)) - t
 
  • #5
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[tex]2t/ sin(t) - t [/tex]
 
  • #6
fzero
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If you mean

[tex]\frac{2t}{\sin t -t},[/tex]

then rewrite it in the form

[tex]\frac{2}{\frac{\sin t}{t} -1}.[/tex]
 
  • #7
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If you factor out the 2 and split the limit into two parts then you have;
2 lim t->0 (t/sint) - lim t->0 (t)
 
  • #8
84
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that was exactly what i was doing wrong.. thank you very much

it becomes

forgive my poor latex.. im trying

[tex]2 / sint/t - t/t[/tex]
sorry ill have to figure out better latex tomorrow
but if im correect

that is 2 / 1 - t/t and since t/t is approaching zero but both equal shouldnt that become 1 as well... giving 1-1 in the denominator and an undefined situation?? the text is saying it should equal DNE or +infinity .... i dont see the infinity situation here?
 
  • #9
fzero
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that was exactly what i was doing wrong.. thank you very much

it becomes

forgive my poor latex.. im trying

[tex]2 / sint/t - t/t[/tex]
sorry ill have to figure out better latex tomorrow
but if im correect

that is 2 / 1 - t/t and since t/t is approaching zero but both equal shouldnt that become 1 as well... giving 1-1 in the denominator and an undefined situation?? the text is saying it should equal DNE or +infinity .... i dont see the infinity situation here?

Well

[tex]\frac{1}{0}=\infty[/tex]

in the sense that

[tex]\lim_{x\rightarrow 0} \frac{1}{x} \rightarrow \infty.[/tex]

So the limit does not exist.
 
  • #10
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but in that form it just purely does not exist am i right? unless you know wether you're approaching zero from the right or the left.. you can end up with -infinity or +infinity....

so it would need to be a one sided limit.. or am i off?
 
  • #11
fzero
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You're right if the left and right limits don't agree, the limit does not exist. However, it's also true that a divergent result means that the limit doesn't exist.

In this case, the function is even, so both the right and left limits give [tex]-\infty[/tex]. We still say that the limit does not exist.
 
  • #12
84
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excellent thanks for your help!... i did look in the text book again to confirm the answer and i see its showing a +infinity?? not sure what the reasoning behind the positive is? but fighting with this question has actually helped me understand a bit deeper. thanks!
 

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