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

[eplymale3043]

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.

 

[Dick]

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

 

[eplymale3043]

it gets close to 1.57~

 

[Dick]

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.

 

[eplymale3043]

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

 

[Dick]

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.

 

[eplymale3043]

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

my calculator was in degrees. ;O

 

[Dick]

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??

 

[eplymale3043]

.0001 ~-9999.999967

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

right?

 

[Dick]

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?

 

[eplymale3043]

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

 

[Dick]

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

 

[eplymale3043]

negative infinity?

 

[Dick]

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

 

[eplymale3043]

0.

as the number decreases, x approaches 0.

so the limit would be 0?

 

[Dick]

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