What is the limit of tan(x)^sin(x) as x->0+

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In summary, the limit of tan(x)^sin(x) as x approaches 0+ is undefined. Various methods, such as L'Hopital's rule, have been suggested to solve the limit, but the final result is still infinity. The conversation also touches upon the use of L'Hopital's rule in limit problems and provides examples of other limits being solved using this method.
  • #1
iatnogpitw
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What is the limit of tan(x)^sin(x) as x-->0+

Homework Statement


[tex]Lim\rightarrow0^{+}((tan(x))^{sin(x)}[/tex]

Homework Equations


I know that you have to raise e to this limit and you can then bring down sin(x) to get [tex](sin(x))(ln(tan(x)))[/tex], but beyond that, I am stuck.

The Attempt at a Solution


I guess above is a bit of both an attempt and relevant equations.
 
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  • #2


Well, [itex]\sin(0)=0[/itex] and [itex]\ln(\tan(0))=\infty[/itex], so If you write it in the form :[tex]\frac{\ln(\tan(x))}{\frac{1}{\sin(x)}}[/tex], you have a limit of the form [tex]\frac{\infty}{\infty}[/tex] and you can apply L'hospital's rule.
 
  • #3


do
lim tan(x)^sin(x)=[lim sin(x)^sin(x)]/[lim cos(x)^sin(x)]
use lim x^x=1

if you want to go forward from where were, do you know l'Hopitals rule?
lim sin(x)log(tan(x))=lim log(tan(x))/csc(x)=lim [log(tan(x))]'/[csc(x)]'
 
  • #4


gabbagabbahey said:
Well, [itex]\sin(0)=0[/itex] and [itex]\ln(\tan(0))=\infty[/itex], so If you write it in the form :[tex]\frac{\ln(\tan(x))}{\frac{1}{\sin(x)}}[/tex], you have a limit of the form [tex]\frac{\infty}{\infty}[/tex] and you can apply L'hospital's rule.

log(tan(0+)=-infinity

Why is L'hospital's rule suggested so frequently on limit problems?
I used to mock the Hughes-Hallett Calculus book for (among other things) omiting L'hospital's rule, but now I am beginning to understand why.

Here is an extra few for practice (all limit's x->0)

lim x/x=lim x'/x'=lim 1/1=lim 1=1
lim x^x=exp(lim [log(x)]'/[1/x]')=exp(lim (1/x)/(-1/x^2))=exp(lim -x)=exp(0)=1
lim [1/x-1/x]=lim 0'/x'=lim 0/1=0
 
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  • #5


Okay, so I got [tex]((1/(tan(x)(cos^{2}(x))))/((sin(x))/(cos^{2}(x)) = (cos(x)/(sin^{2}(x))[/tex] This limit still equals infinity though, and I can' see L'hopital's rule getting me any beyond a term with infinity in it.

EDIT: Oh, so does that mean the limit DNE?
 
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  • #6


iatnogpitw said:
Okay, so I got [tex]((1/(tan(x)(cos^{2}(x))))/((sin(x))/(cos^{2}(x)) = (cos(x)/(sin^{2}(x))[/tex] This limit still equals infinity though, and I can' see L'hopital's rule getting me any beyond a term with infinity in it.

EDIT: Oh, so does that mean the limit DNE?

Errrrmm is [tex]\frac{d}{dx}\frac{1}{\sin x}[/tex] REALLY [tex]\frac{\sin x}{\cos^2 x}[/tex] ?:wink:
 

What is the limit of tan(x)^sin(x) as x->0+?

The limit of tan(x)^sin(x) as x approaches 0 from the positive side is 1. This can be determined by using L'Hopital's rule or by graphing the function.

Why is the limit of tan(x)^sin(x) as x->0+ equal to 1?

The limit of tan(x)^sin(x) as x->0+ is equal to 1 because as x gets closer to 0, both the tangent and sine functions approach 0. This results in tan(x)^sin(x) becoming 1, as any number raised to the power of 0 is equal to 1.

Can the limit of tan(x)^sin(x) as x->0+ be evaluated using other methods?

Yes, the limit of tan(x)^sin(x) as x->0+ can also be evaluated using the squeeze theorem or by using trigonometric identities to rewrite the function. However, using L'Hopital's rule or graphing the function are the most common methods.

Is the limit of tan(x)^sin(x) as x->0+ the same as the limit of sin(x)^tan(x) as x->0+?

No, the limit of sin(x)^tan(x) as x->0+ is undefined, while the limit of tan(x)^sin(x) as x->0+ is 1. This is because sin(x)^tan(x) oscillates between positive and negative infinity as x approaches 0, while tan(x)^sin(x) approaches a constant value of 1.

How can the limit of tan(x)^sin(x) as x->0+ be applied in real-world situations?

The limit of tan(x)^sin(x) as x->0+ can be applied in fields such as physics and engineering to determine the behavior of oscillating systems. It can also be used in economics to model the growth of certain populations or investments. Additionally, it has applications in computer science and signal processing to analyze and predict data patterns.

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