Graphical Illus Sin X Over X

## Description

This interact provides a graphical understanding of why $\frac{sin(x)}{x}$ = 1$

## Sage Cell

#### Code

```
x=var('x')
@interact
def _(x = slider(-7/10,7/10,1/20,1/2)):
pretty_print(html(r'<h3>A graphical illustration of $\lim_{x -> 0} \sin(x)/x =1$</h3>'))
pretty_print(html(r'Below is the unit circle, so the length of the <font color=red>red line</font> is |sin(x)|'))
pretty_print(html(r'and the length of the <font color=blue>blue line</font> is |tan(x)| where x is the length of the arc.'))
pretty_print(html(r'From the picture, we see that |sin(x)| $\le$ |x| $\le$ |tan(x)|.'))
pretty_print(html(r'It follows easily from this that cos(x) $\le$ sin(x)/x $\le$ 1 when x is near 0.'))
pretty_print(html(r'As $\lim_{x ->0} \cos(x) =1$, we conclude that $\lim_{x -> 0} \sin(x)/x =1$.'))
if not (x == 0):
pretty_print("sin(x)/x = "+str(sin(float(x))/float(x)))
elif x == 0:
pretty_print("The limit of sin(x)/x as x tends to 0 is 1.")
C=circle((0,0),1, rgbcolor='black')
mvp = (cos(x),sin(x));tpt = (1, tan(x))
p1 = point(mvp, pointsize=30, rgbcolor='red'); p2 = point((1,0), pointsize=30, rgbcolor='red')
line1 = line([(0,0),tpt], rgbcolor='black'); line2 = line([(cos(x),0),mvp], rgbcolor='red')
line3 = line([(0,0),(1,0)], rgbcolor='black'); line4 = line([(1,0),tpt], rgbcolor='blue')
result = C+p1+p2+line1+line2+line3+line4
result.show(aspect_ratio=1, figsize=[3,3], axes=False)
```

## Options

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Date: 22 Jul 2020 23:42

Submitted by: Zane Corbiere