In the figure,

`h`

is the measure of the angle and also the length of the arc of the circle between `A`

and `D`

(in radians). (I'm using `h`

instead of `x`

because ultimately this will be a small change to `x`

, as in `x + h`

).Kline sets up this inequality:

The last result follows since

`OAB`

and `OED`

are similar triangles so that `ED/OD = AB/OB`

, and `OD = 1`

. Thus, the inequality becomes:As long as we do not actually reach

`h = 0`

, we are allowed to divide by `sin(h)`

:As

`h`

approaches 0, `cos(h)`

and its inverse `1/cos(h)`

both approach 1 so the ratios `h/sin(h)`

and `sin(h)/h`

both approach 1 as well.Strang does the first part by extending the segment AB (dotted line) and noting that

The other thing we need is:

Following Strang:

Since

`sin(h) < h`

(see the figure again):Divide by

`h`

and also by `1 + cos(h`

):Strang says, note that all terms are positive, so that:

Now, as as

`h -> 0`

the right side must go to `0/2`

(that is, `0`

) and our ratio is "caught in the middle."Here is a simple Python script to calculate the same values. Notice that although

`1-cos(h)`

and `h`

are both approaching 0, h is approaching *more slowly*, so the ratio tends to the 0 in the limit. First the output:

and the listing:

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