Epsilon-Delta Limits: Finding the Optimal Delta for a Given Epsilon Value

[ƒ(x)]

Given: limit of (sin x)/x as x --> 0 and that ε = .01
Problem: Find the greatest c such that δ between zero and c is good. Give an approximation to three decimal places.

Equations:

0 < |x - a| < δ
0 < |f(x) - L| < εAttempt:

0 < |x - 0| < δ
0 < | sin(x)/x - 1| < ε

0 < | sin(x)/x - 1| < .01
0 < | sin(x)/x| < 1.01
0 < |sin(x)| < 1.01|x|
0 < |sin(x)| < 1.01δ
0 < |sin(x)|/1.01 < δ
Since sin(x) is going to range between -1 and 1, the greatest value for δ is 1/1.01. But, this answer isn't correct. The correct answer is .245, and I don't know how to get that.

 

[jbunniii]

Given: limit of (sin x)/x as x --> 0 and that ε = .01
Problem: Find the greatest c such that δ between zero and c is good. Give an approximation to three decimal places.

What do you mean by "δ between zero and c is good"?

My guess is that you mean:

"Find the greatest $c$ such that

$$\left|\frac{\sin x}{x} - 1 \right| < \epsilon$$

whenever

$$0 < |x - 0| < \delta \leq c$$."

Is that right?

[It seems clearer to dispense with $c$ and simply ask "how big can $\delta$ be?"]

Your problem is the following step:

Equations:

0 < | sin(x)/x - 1| < .01
0 < | sin(x)/x| < 1.01

While the upper bound in the second line is true, it doesn't buy you anything. In fact, it's true no matter what $\epsilon$ is, because $$|\sin(x)/x|$$ never exceeds 1.

Also, the lower bounds are wrong: both $$|\sin(x)/x - 1|$$ and $$|\sin(x)/x|$$ do equal 0 for some values of $x$. We don't want/need to disallow that possibility.

Try writing

$$\left|\frac{\sin x}{x} - 1\right| < 0.01$$

in the following equivalent way

$$-0.01 < \frac{\sin x}{x} - 1 < 0.01$$

and thus

$$0.99 < \frac{\sin x}{x} < 1.01$$

The right-hand inequality is true for all $x$, so we only need to consider the left inequality:

$$0.99 < \frac{\sin x}{x}$$

Now you need to find what range of $x$ satisfies this inequality. I assume you are permitted to do so numerically (using a computer or calculator)?

 

[ƒ(x)]

Yes, that works. Thank you.

 

[jbunniii]

Oops, for the record, I just noticed that I lied when I said that $$|\sin(x)/x - 1|$$ can equal 0. It can't. But my point that you don't need the $$0 < |\sin(x)/x - 1|$$ part of the inequality is still correct.

 

[ƒ(x)]

Ok. :)