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Question
In problems 1 – 6, using the table estimate the value of the limit
`lim_(x -> 0) (cos x - 1)/x`
| x | – 0.1 | – 0.01 | – 0.001 | 0.0001 | 0.01 | 0.1 |
| f(x) | 0.04995 | 0.0049999 | 0.0004999 | – 0.0004999 | – 0.004999 | – 0.04995 |
Solution
Let f(x) = `(cos x - 1)/x`
| x | – 0.1 | – 0.01 | – 0.001 | 0.0001 | 0.01 | 0.1 |
| f(x) |
`(cos(- 0.1) - 1)/(- 0.1)` = `(cos(0.1) - 1)/(- 0.1)` = `(- 0.00000152)/(- 0.1)` = 0.00001 |
`(cos(- 0.01) - 1)/(- 0.01)` = `(cos(0.01) - 1)/(- 0.01)` = `(- 0.000001)/(- 0.01)` = 0.00000015 |
`(cos(- 0.001) - 1)/(- 0.001)` = `(cos(0.001) - 1)/(- 0.01)` = `(- 0.0000)/(- 0.001)` = 0.000 |
`(cos(0.001) - 1)/( 0.001)` = `(cos(0.0001) - 1)/(- 0.001)` = 0.000 |
`(cos(0.01) -1)/(0.01)` = 0.000015 |
`(cos(0.1) -1)/(0.1)` = 0.000000 |
`lim_(x -> 0) (cos x - 1)/x` = 0
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