Limites : limit

### 2.19.1 Limites : `limit`

`limit` compute the limit of an expression in a finite or infinite point. It is also possible with an optional argument to compute unidirectional limit (1 for right limit and -1 for left limit .
`limit` takes three or four arguments :
an expression, the name of a variable (for example `x`), the limit point (for example `a`) and an optional argument, by default `0`, to indicate if the limit is unidirectional. This argument is equal to `-1` for a left limit (`x<a`) or is equal to `1` for a right limit (`x>a`) or is equal to `0` for a limite.
`limit` returns the limit of the expression when the variable (for example `x`) approachs the limit point (for example `a`).
Remark
It is also possible to put `x=a` as argument instead of `x,a`, hence : `limit` takes also as arguments an expression depending of a variable, an equality (variable =value of the limit point) and perhaps 1 or -1 to indicate the direction.
Input :

`limit(1/x,x,0,-1)`

or

`limit(1/x,x=0,-1)`

Output :

`-(infinity)`

Input :

`limit(1/x,x,0,1)`

or

`limit(1/x,x=0,1)`

Output :

`+(infinity)`

Input :

`limit(1/x,x,0,0)`

or

`limit(1/x,x,0)`

or

`limit(1/x,x=0)`

Output :

`infinity`

Hence, `abs(1/x)` approachs +∞ when x approachs 0.

Exercises :

- Find for n>2, the limit when x approachs 0 of :

  ntan(x)−tan(nx)
  sin(nx)−nsin(x)

  Input :
  `limit((n*tan(x)-tan(n*x))/(sin(n*x)-n*sin(x)),x=0)`
  Output :
  `2`
- Find the limit when x approachs +∞ of :

  √
  x+ |
  √
  x+ | √ |
  x
  | − | √ |
  x

  Input :
  `limit(sqrt(x+sqrt(x+sqrt(x)))-sqrt(x),x=+infinity)`
  Output :
  `1/2`
- Find the limit when x approachs 0 of :

  √ |
  1+x+x2/2
  | −exp(x/2)
  (1−cos(x))sin(x)

  Input :
  `limit((sqrt(1+x+x```^```2/2)-exp(x/2))/((1-cos(x))*sin(x)),x,0)`
  Output :
  `-1/6`

Remark
To compute limit, it is better sometimes to quote the first argument.
Input :

`limit(’(2*x-1)*exp(1/(x-1))’,x=+infinity)`

Note that the first argument is quoted, because it is better that this argument is not simplified (i.e. not evaluated).
Output :

`+(infinity)`