What is the attempt at a solution for substituting pi/14?

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In summary, the conversation discusses the behavior of the function tan(x) as x approaches pi/2 from above. It is observed that as x gets closer to pi/2, the value of tan(x) approaches infinity. The conversation also explores the behavior of e^(tan(7x)) as x approaches pi/2 from above, and it is determined that the limit of this function is 0. The conversation also touches on the topic of calculator approximations and the importance of using precise values in calculations.
  • #1
eplymale3043
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Homework Statement



http://i38.tinypic.com/v6m1bo.png

Homework Equations





The Attempt at a Solution



I tried substituting pi/14 it doesn't accept that answer.
 
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  • #2


7*pi/14=pi/2. What's the behavior of tan(x) like as x->pi/2 from above?
 
  • #3


it gets close to 1.57~
 
  • #4


7*x gets close to pi/2=1.57~, as x->(pi/14)+, sure. But x->(pi/14)+ means thinking of x's a little larger than pi/14. What's tan(7*(pi/14+0.01)), tan(7*(pi/14+0.001)) etc etc. Punch them into a calculator if you have to.
 
  • #5


tan(7 ( pi / 14 +.01 ) ) is .028645
tan(7 ( pi / 14 +.001 ) ) is .027545
tan(7 ( pi / 14 +.0001 ) ) is .027435
tan(7 ( pi / 14 +.00001 ) ) is .027424

so basically, its approaching .02742
 
  • #6


I get tan(7*((pi/14)+0.001))=-1428.57... Are you using an bad approximation of pi? Don't. 7*pi/14+(a small number) is pi/2+(a small number). What's tan(pi/2+(a small number))? Look at a graph of tan.
 
  • #7


I got -999.9996667 for tan(pi/2+.001)

my calculator was in degrees. ;O
 
  • #8


eplymale3043 said:
I got -999.9996667 for tan(pi/2+.001)

my calculator was in degrees. ;O

I forgot about that way of making a mistake. Good for you for catching it. Now if 0.001 gets even smaller??
 
  • #9


.0001 ~-9999.999967

but when I use .00001 my calculator rounds to -100000 but it should be around -99999.9999967

right?
 
  • #10


Sure, sure. But the point is that tan(7x) is going to minus infinity as x->(pi/14)+, right? Again look at a graph of tan(x) just above pi/2. Agreed?
 
  • #11


Yes, I see it is approaching minus infinity, I can also see it on the graph as well.
 
  • #12


Good, good. So what does e^(tan(7*x)) approach?
 
  • #13


negative infinity?
 
  • #14


tan(7x) approaches negative infinity. What's, oh, say e^(-100000000)? If your calculator overflows, try e^(-100).
 
  • #15


0.

as the number decreases, x approaches 0.

so the limit would be 0?
 
  • #16


You said it, not me. But x doesn't approach 0. e^tan(7*x) approaches 0 as x->(pi/14)+. Sure, 0.
 

What is the definition of a limit?

A limit is the value that a function or sequence approaches as the input or variable approaches a certain value. It is a fundamental concept in calculus and is used to describe the behavior of a function near a specific point.

How do you find the limit of a function?

The limit of a function can be found by evaluating the function as the input approaches the desired value. This can be done algebraically, graphically, or numerically using a table of values. If the function is continuous at the desired value, the limit is equal to the value of the function at that point.

What is the significance of pi/14 in limits?

Pi/14 is a commonly used value in limits because it is a rational multiple of pi (a transcendental number) and can be easily substituted into trigonometric functions. It is also a useful value when working with angles and radians.

Can pi/14 be substituted into any function?

No, pi/14 can only be substituted into functions that involve trigonometric functions since it is a value related to angles. It cannot be substituted into polynomial or exponential functions, for example.

What is the purpose of substituting pi/14 into a limit?

Substituting pi/14 into a limit allows for the evaluation of the limit at a specific point. This can help determine the behavior of the function at that point and can be used to find the value of the function at that point if it is continuous. It is also useful in solving more complex problems involving trigonometric functions.

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