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5.19.1  Limits : limit

limit computes the limit of an expression at a finite or infinite point. It is also possible with an optional argument to compute a one-sided limit (1 for the right limit and -1 for the 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 limit.
limit returns the limit of the expression when the variable (for example x) approaches 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) approaches +∞ when x approaches 0.

Exercises :

Remark
To compute limits, 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)

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