With an example, calculation of limits for a function with special property is discussed.

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Given function `f(x)=|x-1|`. what is `f(x)|_(x=1)`?

- `0`
- `0`
- `-1`
- `+1`

The answer is '`0`'

Given function `f(x)=|x-1|`. what is left-hand-limit `lim_(x->1-)f(x)`?

- `0`
- `0`
- `-1`
- `+1`

The answer is '`0`'

The left-hand-limit `lim_(x->1-)f(x)`

`quad quad = |1-delta-1|`

`quad quad = |-delta|`

`quad quad = delta`

`quad quad = 0 quad quad quad quad` substituting `delta=0`

Given function `f(x)=|x-1|`. what is right-hand-limit `lim_(x->1+)f(x)`?

- `0`
- `0`
- `-1`
- `+1`

The answer is '`0`'.

The right-hand-limit `lim_(x->1+)f(x)`

`quad quad = |1+delta-1|`

`quad quad = |delta|`

`quad quad = delta`

`quad quad = 0 quad quad quad quad` substituting `delta=0`

Given function `f(x)=|x-1|`.

• `f(1)=0`

• `lim_(x->1-)f(x) = 0`

• `lim_(x->1+)f(x) = 0`

The function is continuous at `x=1`. Note that the function undergoes an abrupt change at `x=1`.

• for `x<1`, the slope of the line is `-1`

• for `x>1`, the slope of the line is `1`

• At `x=1`, the slope of the line is not defined.

*Solved Exercise Problem: *

Given the function `f(x)=|sin x|`. Is the function continuous or defined at `x=pi`?

- continuous
- defined and not continuous
- defined and not continuous
- not defined and not continuous

The answer is 'continuous'

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