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Concept of Infinity

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Concept of Infinity - Lesson Summary

The nature of the function f(x) = 1 x

lim x → 0 + f(x)  = lim x → 0 + 1 x

    x

  1

  0.5

  0.2

  0.1=10-1

  0.01=10-2

  …

  10-n

  f(x)

  1

  2

  5

  10 = 101

  100 = =102

  …

  10n

⇒ For a positive real number very close to 0, the value of the function will be a large number.

lim x → 0 + f(x)  = lim x → 0 + 1 x = + ∞

⇒ Right hand limit of f(x) at 0 does not exist.

lim x → 0 – f(x)  = lim x → 0 – 1 x

  x

  - 1

  - 0.5

  - 0.2

  -10-1

  10-2

  …

  -10-n

  f(x)

  - 1

  - 2

  - 5

  -101

  -102

  …

  -10n

⇒ For a negative real number very close to 0, the value of the function will be a very small number.


lim x → 0 + f(x)  = lim x → 0 + 1 x = – ∞

⇒ Left hand limit of f(x) at 0 does not exist.

The nature of the function f(x) = tan x

  xo

  tan xo

  0°

  0

  45°

  1.0000

  60°

  1.7320

  85°

  11.4300

  89°

  57.2899

  89.9°

  572.9572

  89.99°

  5729.5778

  89.999°

  57295.7795

  89.9999°

  572957.7951

  90°

  Infinity

⇒ For an angle very close to 90° from the left side, the value of the function will be a large number.

This number is represented by + ∞.

lim x → 90 ∘  - f(x)   = lim x → 90 ∘  - tan x   = + ∞

  xo

  tan xo

  180°

      0

  150°

  - 0.5773

  120°

  - 1.7320

  100°

  - 5.6712

  91°

  - 57.2899

  90.1°

  - 572.9572

  90.01°

  - 5729.5778

  90.001°

  - 57295.7795

  90.0001°

  - 572957.7951

  90°

  Infinity

⇒ For an angle very close to 90° from the right side, the value of the function will be a small number. This number is represented by – .

lim x → 90 ∘  + f( x )  = lim x → 90 ∘  + tan x   = – ∞

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