Understanding Limits: Delta-Epsilon Proof Practice Problems and Solutions

[the_quack]

I am doing these practice problems to study for a test and I just can't figure them out, it's kind of frustrating.

The first concerns the delta-epsilon proof.

f(x) = 2-(1/x)
x is approaching 1, and the limit is 1.
Epsilon is given at 0.1

so, what I have...
0<|x-1|<d (delta)
|f(x)-1|<.1
so
|2-(1/x)-1|<.1
|1-(1/x)|<.1

Maybe I am just an idiot, but how do I go from there?
The answer is delta=1/11, I just can't see how to get it...

I mean, I have done other problems like it,
like for lim x->3 (2x-5) L=1 e=.01
|2x-5-1|<.01
2|x-3|<.01
|x-3|<.005 so delta=.005

And this other problem:
find the limit, using algebra, for x->0 of sin7x/sin9x

I multiplied both sides by csc9x, to get (sin7x*csc9x)/1
because sinx*cscx=1, sin9x*csc9x=1

But I can figure out where to go from there.
Am I doing it wrong?

Thank you for any help!

 

[the_quack]

Sorry to bother you guys, but can anyone explain to me how to find the standard equation of the tangent line to f(x) parallel to a given line? (so slope is known)

I desperately need to know how to do this particular thing...

Thanks!

 

[HallsofIvy]

I am doing these practice problems to study for a test and I just can't figure them out, it's kind of frustrating.

The first concerns the delta-epsilon proof.

f(x) = 2-(1/x)
x is approaching 1, and the limit is 1.
Epsilon is given at 0.1

so, what I have...
0<|x-1|<d (delta)
|f(x)-1|<.1
so
|2-(1/x)-1|<.1
|1-(1/x)|<.1

Maybe I am just an idiot, but how do I go from there?
The answer is delta=1/11, I just can't see how to get it...

Okay, so $|\frac{x-1}{x}|< .1$
You need to make |x-1|< 0.1|x|.
If x is close to 1, how small can x be? How small must |x-1| be?

I mean, I have done other problems like it,
like for lim x->3 (2x-5) L=1 e=.01
|2x-5-1|<.01
2|x-3|<.01
|x-3|<.005 so delta=.005

And this other problem:
find the limit, using algebra, for x->0 of sin7x/sin9x

I multiplied both sides by csc9x, to get (sin7x*csc9x)/1
because sinx*cscx=1, sin9x*csc9x=1

But I can figure out where to go from there.
Am I doing it wrong?

Thank you for any help!

Multiplied both sides of what by csc 9x? Looks to me like you just replaced 1/sin 9x by csc 9x.

Anyway, do you know the limit of sin x/ x as x goes to 0?

What would you do to get sin ax/x into that form?