With an example, calculation of limits for a continuous function is discussed. The condition under which a function is continuous is illustrated with examples.

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Given function `f(x)=1/(x^2+1)`. what is `f(x)|_(x=1)`?

- Substitute `x=1`
- `1/(1+1)`
- `1/2`
- all the above
- all the above

The answer is 'All the above'. The options provide the steps to evaluate `f(1)`.

Given function `f(x)=1/(x^2+1)`. what is left-hand-limit `lim_(x->1-)f(x)`?

- Substitute `x=1`
- Substitute `x=1+delta`
- Substitute `x=1-delta`
- Substitute `x=1-delta`
- all the above

The answer is 'Substitute `x=1-delta`'.

`lim_(x->1-)f(x)`

`quad quad = 1/((1-delta)^2+1)`

`quad quad = 1/(1-2delta+delta^2+1)`

`quad quad = 1/(2-2delta+delta^2)`

`quad quad = 1/2 quad quad quad quad` (substituting `delta=0`)

Given function `f(x)=1/(x^2+1)`. what is right-hand-limit `lim_(x->1+)f(x)`?

- Substitute `x=1`
- Substitute `x=1+delta`
- Substitute `x=1+delta`
- Substitute `x=1-delta`
- all the above

The answer is 'Substitute `x=1+delta`'.

`lim_(x->1+)f(x)`

`quad quad = 1/((1+delta)^2+1)`

`quad quad = 1/(1+2delta+delta^2+1)`

`quad quad = 1/(2+2delta+delta^2)`

`quad quad = 1/2 quad quad quad quad` (substituting `delta=0`)

Given function `f(x)=1/(x^2+1)`.

• `f(1)=1/2`

• `lim_(x->1-)f(x) = 1/2`

• `lim_(x->1+)f(x) = 1/2`

The function is continuous at `x=1`.

Note: Though, as part of the topic, the functions that are discontinuous or indeterminate or non-differentiable are mainly handled, finding limits is not only for those kind of functions. Any function at any point can be evaluated for limits.

Function is *continuous* if all three values are equal.

**Function is continuous: ** if `f(a)` `= lim_(x->a-)f(x)` `=lim_(x->a+)f(x)`, then the function is continuous at `x=a`.

*Solved Exercise Problem: *

Given function `f(x) = sin x`, is it continuous at input values `x= 0, pi/2, pi`?

- Continuous at `0` and `pi/2` only
- Continuous at all three input values
- Continuous at all three input values
- Continuous at only `0`
- not continuous at any one of the given values

The answer is 'Continuous at all three input values'

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