Math3-S2-Demo2_limits.mw

Math 3 Winter 2004

Introduction to Calculus


Class Demo for the Limits of Functions

January 7, 2004


   We first consider lim sin(x)/x   as proc (x) options operator, arrow; 0 end proc .

> f := x -> sin(x)/x:

> f(0.01);

.9999833334

> printf("            x               f(x)\n"):
for n from 0 to 5 do
x := 1 / 2^n:
printf("  %15.10f   %15.10f\n", x, f(x));
od:

           x               f(x)
    1.0000000000      0.8414709848
    0.5000000000      0.9588510772
    0.2500000000      0.9896158372
    0.1250000000      0.9973978672
    0.0625000000      0.9993490854
    0.0312500000      0.9998372477

    Observe this on the graph

> plot(f, -Pi..Pi);

[Plot]

    Another Limit

    Consider lim (sqrt(t^2+9)-3)/t^2 as proc (t) options operator, arrow; 0 end proc .

> g := t -> (sqrt(t^2 + 9) - 3) / t^2:

> g(0.5);

.1655250600

> printf("            t             g(t)\n"):
for n from 0 to 5 do
t := 1 / 2^n:
printf("  %15.10f   %15.10f\n", t, g(t));
od:

           t             g(t)
    1.0000000000      0.1622776600
    0.5000000000      0.1655250600
    0.2500000000      0.1663783200
    0.1250000000      0.1665944000
    0.0625000000      0.1666486000
    0.0312500000      0.1666620000

     As always, lets look at a picture.

> plot(g, -1..1);

[Plot]

> plot(g, -1e-5..1e-5, 0.16..0.17);

[Plot]

> Digits := 50:

  ... and another one

     Consider lim sin Pi/x   as proc (x) options operator, arrow; 0 end proc .

> h := x -> sin(Pi / x):

> printf("            x             h(x)\n"):
for n from 0 to 5 do
x := 1 / 2^n:
printf("  %15.10f   %15.10f\n", x, h(x));
od:

           x             h(x)
    1.0000000000      0.0000000000
    0.5000000000      0.0000000000
    0.2500000000      0.0000000000
    0.1250000000      0.0000000000
    0.0625000000      0.0000000000
    0.0312500000      0.0000000000

> plot(h, -1..1);

[Plot]