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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 â†’ 0 + f(x)  = lim x â†’ 0 + 1 x

x

1

0.5

0.2

0.1=10-1

0.01=10-2

â€¦

10-n

f(x)

1

2

5

10 = 101

100 = =102

â€¦

10n

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

lim x â†’ 0 + f(x)  = lim x â†’ 0 + 1 x = + âˆž

â‡’ Right hand limit of f(x) at 0 does not exist.

lim x â†’ 0 â€“ f(x)  = lim x â†’ 0 â€“ 1 x

x

- 1

- 0.5

- 0.2

-10-1

10-2

â€¦

-10-n

f(x)

- 1

- 2

- 5

-101

-102

â€¦

-10n

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

lim x â†’ 0 + f(x)  = lim x â†’ 0 + 1 x = â€“ âˆž

â‡’ 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

â‡’ 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 + âˆž.

lim x â†’ 90 âˆ˜  - f(x)   = lim x â†’ 90 âˆ˜  - tan x   = + âˆž

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

â‡’ 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 â€“ âˆž.

lim x â†’ 90 âˆ˜  + f( x )  = lim x â†’ 90 âˆ˜  + tan x   = â€“ âˆž