Notes On Concept of Infinity - CBSE Class 12 Maths

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   = –

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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