Precise Definition Limit

## Description

This interact allows the user to see and experiment with the delta-epsilon definition of a limit. Instructions are provided in the interact itself.

## Sage Cell

#### Code

pretty_print(html("<h2>Limits: <i>ε-δ</i></h2>"))
pretty_print(html("This allows you to estimate which values of <i>δ</i> guarantee that <i>f</i> is within <i>ε</i> units of a limit."))
pretty_print(html("<ul><li>Modify the value of <i>f</i> to choose a function.</li>"))
pretty_print(html("<li>Modify the value of <i>a</i> to change the <i>x</i>-value where the limit is being estimated.</li>"))
pretty_print(html("<li>Modify the value of <i>L</i> to change your guess of the limit.</li>"))
pretty_print(html("<li>Modify the values of <i>δ</i> and <i>ε</i> to modify the rectangle.</li></ul>"))
pretty_print(html("If the blue curve passes through the pink boxes, your values for <i>δ</i> and/or <i>ε</i> are probably wrong."))
@interact
def delta_epsilon(f = input_box(default=(x^2-x)/(x-1), label="$f$"), a=input_box(default=1, label="$a$"), L = input_box(default=1, label="$L$"), delta=input_box(label=r"$\delta$",default=0.1), epsilon=input_box(label=r"$\varepsilon$",default=0.1), xm=input_box(label=r"$x_{min}$",default=-1), xM=input_box(label=r"$x_{max}$",default=4)):
f_left_plot = plot(f,xm,a-delta/3,thickness=2)
f_right_plot = plot(f,a+delta/3,xM,thickness=2)
epsilon_line_1 = line([(xm,L-epsilon),(xM,L-epsilon)], rgbcolor=(0.5,0.5,0.5),linestyle='--')
epsilon_line_2 = line([(xm,L+epsilon),(xM,L+epsilon)], rgbcolor=(0.5,0.5,0.5),linestyle='--')
ym = min(f_right_plot.ymin(),f_left_plot.ymin())
yM = max(f_right_plot.ymax(),f_left_plot.ymax())
aL_point = point((a,L),rgbcolor=(1,0,0),pointsize=20)
delta_line_1 = line([(a-delta,ym),(a-delta,yM)],rgbcolor=(0.5,0.5,0.5),linestyle='--')
delta_line_2 = line([(a+delta,ym),(a+delta,yM)],rgbcolor=(0.5,0.5,0.5),linestyle='--')
(f_left_plot +f_right_plot +epsilon_line_1 +epsilon_line_2 +delta_line_1 +delta_line_2 +aL_point +bad_region_1 +bad_region_2).show(xmin=xm,xmax=xM)


none

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