Solving a Limit Involving tanx and pi/4

  • Thread starter Glissando
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Do you see how this is done?In summary, the problem is finding the limit of (tanx-1)/(x-pi/4) as x approaches pi/4. This can be rewritten using the formula tanx = sinx/cosx and simplified to [(sinx-cosx)/cosx]/(x-pi/4). From here, we can use the formula tan(x)=tan((x-\pi/4)+\pi/4) and apply this to simplify the expression further in terms of x-\pi/4. The goal is to write everything in function of x-\pi/4.
  • #1
Glissando
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Homework Statement


Find the limit:

lim (tanx-1)/(x-pi/4)
x->pi/4


Homework Equations


tanx = sinx/cosx


The Attempt at a Solution



lim (sinx/cosx-1)/(x-pi/4)
x->pi/4

lim [(sinx-cosx)/cosx]/(x-pi/4)
x->pi/4

I have no idea what to do after this ): I also tried squaring the whole function and getting tan2x-1 = sec2x, but I get so lost on what to do with pi/4 because it just keeps becoming undefined!

Thanks for your help!
 
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  • #2
Hi Glissando! :smile:

I take it you're not allowed to use L'hopitals rule?? In that case, I would first write

[tex]\tan(x)=\tan((x-\pi/4)+\pi/4)[/tex]

and work that out. That way you can write everything in function of [itex]x-\pi/4[/itex].
 
  • #3
micromass said:
Hi Glissando! :smile:

I take it you're not allowed to use L'hopitals rule?? In that case, I would first write

[tex]\tan(x)=\tan((x-\pi/4)+\pi/4)[/tex]

and work that out. That way you can write everything in function of [itex]x-\pi/4[/itex].

Hi micromass,

Thanks for your quick response! I'm not too sure what you mean by working it out...do I plug that back into the original equation? Am I solving for x?

Thanks!
 
  • #4
Well, you know formula's for [itex]\tan(\alpha+\beta)[/itex]. So apply these formula's on

[tex]\tan((x-\pi/4)+\pi/4)[/tex]

Our goal is to write everything in function of [itex]x-\pi/4[/itex]
 

Related to Solving a Limit Involving tanx and pi/4

1. What is a limit involving tanx and pi/4?

A limit involving tanx and pi/4 is a mathematical concept that involves finding the value that a function approaches as the input (x) approaches a specific value (pi/4). In this case, the function is tangent (tanx) and the specific value is pi/4.

2. How do you solve a limit involving tanx and pi/4?

To solve a limit involving tanx and pi/4, you can use trigonometric identities and algebraic manipulation to rewrite the function in a form that allows you to directly substitute the specific value (pi/4) for x. You can then evaluate the limit by plugging in the value and simplifying the expression.

3. What is the importance of solving a limit involving tanx and pi/4?

Solving a limit involving tanx and pi/4 is important because it helps us understand the behavior of the tangent function near the specific value of pi/4. This can be useful in various applications, such as calculating slopes and angles in trigonometry and physics problems.

4. Is there a specific method for solving a limit involving tanx and pi/4?

Yes, there are several methods that can be used to solve a limit involving tanx and pi/4, including the squeeze theorem, L'Hospital's rule, and trigonometric identities. The best method to use will depend on the specific function and limit being evaluated.

5. What are some common mistakes to avoid when solving a limit involving tanx and pi/4?

Some common mistakes to avoid when solving a limit involving tanx and pi/4 include forgetting to use trigonometric identities to rewrite the function, incorrectly applying L'Hospital's rule, and not checking for any potential discontinuities or undefined values at the specific value (pi/4). It is also important to carefully simplify the expression before plugging in the value to avoid any errors.

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