Exploring the Effect of Large δ on Sinusoidal Function at x=1.5

[dE_logics]

In a sinusoidal function...suppose the value of δ is very large...then as x approaches any a, the value of f(x) might not approach L directly...or there should not be a direct relation; example -

$$\lim_{x \to 1.5} sin x = 0.997494986$$

Where I've stated δ as 7...then if x = 1.5 – 6.9 = -5.4; as x approach 1.5 from -5.4, value of sin x does not directly approach 0.997494986...it fluctuates between 1 to -1 many times before it reaches that value.

My question is...is this expression $$\lim_{x \to 1.5} sin x = 0.997494986$$ with δ as 7 valid?

 

[HallsofIvy]

It does not matter "how" x approaches a. The only requirement is that "if |x-a|< delta, then |f(x)- L|< epsilon. It is NOT a matter of x getting "closer and closer to a".

Talking about f(x) changing "as x approaches 1.5", for x distant from 1.5 is completely irrelevant. Given any epsilon> 0, there exist a delta such that if |x- 1.5|< delta, then |sin(x)-0.5381|< epsilon.

 

[dE_logics]

It does not matter "how" x approaches a. The only requirement is that "if |x-a|< delta, then |f(x)- L|< epsilon. It is NOT a matter of x getting "closer and closer to a".

Talking about f(x) changing "as x approaches 1.5", for x distant from 1.5 is completely irrelevant. Given any epsilon> 0, there exist a delta such that if |x- 1.5|< delta, then |sin(x)-0.5381|< epsilon.

Oh, ok, I get it...I think.

|sin(x)-0.5381| should not exceed ε if |x- 1.5|< delta.

 

[truth is life]

Oh, ok, I get it...I think.

|sin(x)-0.5381| should not exceed ε if |x- 1.5|< delta.

Rather the other way around. If |x-1.5| < delta, then |sin(x)-.05381| will be less than epsilon. That's the point of the delta-epsilon proof.

 

[dE_logics]

We can take either ways.

 

[emyt]

We can take either ways.

actually, watch out for the false definition:

for any epsilon > 0, there exists a delta > 0 such that |f(x) - L | < epsilon => |x-a| < delta

this is WRONG. it would be a good exercise disproving this

 

[Moo Of Doom]

"B if A" is the same as "if A, then B." If you read carefully, you'll notice dE_logics said the right thing (except with an incorrect value for the limit. I don't know where Halls got 0.5381 from...).

 

[HallsofIvy]

"B if A" is the same as "if A, then B." If you read carefully, you'll notice dE_logics said the right thing (except with an incorrect value for the limit. I don't know where Halls got 0.5381 from...).

Neither do I! I don't know where I got that.

 

[emyt]

"B if A" is the same as "if A, then B." If you read carefully, you'll notice dE_logics said the right thing (except with an incorrect value for the limit. I don't know where Halls got 0.5381 from...).

yeah, I noticed that, but it is good practise to disprove the false statement anyway, many functions work under that particular kind of false definition