Limit of (tan(x))^(tan(2x)) as x approaches pi/4

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In summary: You should try to find the limit of the log of that expression, as was suggested in the previous post. In summary, the limit of the given expression \frac{Lim}{x-> \pi/4} tan(x)^{tan(2x)} does exist and is equal to \frac{1}{e}, as shown by using logarithms and setting both sides to the base e. This method helps simplify the problem and find the limit more easily.
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Homework Statement


[itex]\frac{Lim}{x-> \pi/4}[/itex] tan(x)[itex]^{tan(2x)}[/itex]

[itex]\frac{Lim}{x-> \pi/4}tan(2x) [/itex] does not exist.
However, Wolfram Alpha and my TI-89 say that [itex]\frac{Lim}{x-> \pi/4}[/itex] tan(x)[itex]^{tan(2x)}[/itex]does exist, and that it's [itex]\frac{1}{e}[/itex]
I submitted this answer (it's web based homework and calculators are allowed) and found it was correct. I still don't understand how though.

Homework Equations


tan(2x) = [itex]\frac{2tan(x)}{1-tan^{2}(x)}[/itex]

The Attempt at a Solution


I attempted to split it in to
[itex] \frac{sin(x)^{tan(2x)}}{cos(x)^{tan(2x)}} [/itex], and then use L'Hospital's rule, but I can't seem to get the tan(2x) to go away. Both appear to be indeterminate, but neither one is 0 at the same time, so it doesn't appear that I should be using L'Hopital's rule in this case. However, I can't see any other way to proceed.

Thanks in advance.
 
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  • #2


Thanks.

So people can't do these tasks without calculators anymore? :-)
Hey, at the very least my professor doesn't allow them on tests. :approve:
 
  • #3
Have you tried using

[tex]\tan(2x) = \frac{2 \tan(x)}{1 - \tan^2(x)}[/tex]
 
  • #4
jbunniii said:
Have you tried using

[tex]\tan(2x) = \frac{2 \tan(x)}{1 - \tan^2(x)}[/tex]

There is the good response here, and I particularly agree with this because this generally makes the simplification a bit simple.

Don't forget that you have the function as the exponent of the another function! Here is the hint:

Let y = lim x→π/4 (tan(x))^(tan(2x)). Then, perform logarithms, and we have...

ln(y) = lim x→π/4 tan(2x) * ln(tan(x))

Mod note: Removed intermediate steps students should work out on their own.

Also don't forget to set both sides by e. You should get the results. Let me know if this helps.

Key: ♪ Practice, practice! You will get better with limits like this! ♫
 
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  • #5
Same-same said:

Homework Statement


[itex]\frac{Lim}{x-> \pi/4}[/itex] tan(x)[itex]^{tan(2x)}[/itex]

[itex]\frac{Lim}{x-> \pi/4}tan(2x) [/itex] does not exist.
However, Wolfram Alpha and my TI-89 say that [itex]\frac{Lim}{x-> \pi/4}[/itex] tan(x)[itex]^{tan(2x)}[/itex]does exist, and that it's [itex]\frac{1}{e}[/itex]
I submitted this answer (it's web based homework and calculators are allowed) and found it was correct. I still don't understand how though.

Homework Equations


tan(2x) = [itex]\frac{2tan(x)}{1-tan^{2}(x)}[/itex]

The Attempt at a Solution


I attempted to split it in to
[itex] \frac{sin(x)^{tan(2x)}}{cos(x)^{tan(2x)}} [/itex], and then use L'Hospital's rule, but I can't seem to get the tan(2x) to go away. Both appear to be indeterminate, but neither one is 0 at the same time, so it doesn't appear that I should be using L'Hopital's rule in this case. However, I can't see any other way to proceed.

Thanks in advance.
Yes, it's true that[itex]\displaystyle \lim_{x\to \pi/4} \tan(2x)[/itex] does not exist.

But [itex]\displaystyle \lim_{x\to \pi/4} \tan(x)=1\,,[/itex] and [itex]\displaystyle \lim_{x\to (\pi/4)^+} \tan(2x)=+\infty\ .[/itex]


Find the limit of the log of that expression.

If [itex]\displaystyle \lim_{x\to \pi/4} \ln\left(\left(\tan(x)\right)^{\tan(2x)}\right)=L\,,[/itex]

then [itex]\displaystyle \lim_{x\to \pi/4} \left(\tan(x)\right)^{\tan(2x)}=e^L\ .[/itex]
 
Last edited:

1. What does the limit of (tan(x))^(tan(2x)) as x approaches pi/4 approach?

The limit of (tan(x))^(tan(2x)) as x approaches pi/4 approaches 1.

2. How do you find the limit of (tan(x))^(tan(2x)) as x approaches pi/4?

To find the limit, substitute pi/4 for x in the expression (tan(x))^(tan(2x)) and simplify the resulting expression.

3. Can the limit of (tan(x))^(tan(2x)) as x approaches pi/4 be evaluated using L'Hopital's rule?

Yes, L'Hopital's rule can be used to evaluate the limit, as both the numerator and denominator approach 0 as x approaches pi/4.

4. What is the significance of the limit of (tan(x))^(tan(2x)) as x approaches pi/4?

The limit represents the behavior of the function as x approaches a critical point, in this case pi/4. It can provide insight into the behavior of the function at that point and its overall behavior.

5. Can the limit of (tan(x))^(tan(2x)) as x approaches pi/4 be evaluated using a graphing calculator?

Yes, a graphing calculator can be used to evaluate the limit by graphing the function and observing the behavior of the graph as x approaches pi/4.

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