13.1.1  Univariate function limits

The limit command computes limits, both at numbers and infinities, and in the real case it can compute one-sided limits.

• limit takes three mandatory and one optional argument.
  • expr, an expression.
  • x, the name of a variable.
  • a, the limit point.
  • Optionally, side (either 0, −1 or 1), to specify which side to take a one-sided limit (by default side=0).
  Alternatively, one-sided limits may be specified by using symbols left and right instead of −1 and 1.
• limit(expr,x,a ⟨,side ⟩) returns the limit of expr as x approaches a.
  • If side=0 (the default), then the ordinary limit is returned.
  • If side=−1 or side=left, then the limit from the left (x<a) is returned.
  • If side=1 or side=right, then the limit from the right (x>a) is returned.
• It is also possible to put x=a as argument instead of x,a; limit(expr,var=pt ⟨,side ⟩) is equivalent to limit(expr,var,pt ⟨,side ⟩).

Examples

To find lim_{x→ 0⁻}1/x, input:

or:

To find lim_{x→ 0⁺}1/x, input:

or:

To find lim_{x→ 0}1/x, input:

or:

or:

Note that ∞ or infinity without an explicit + or - represents unsigned infinity.

Find, for n>2, the limit of ntan(x)−tan(nx)/sin(nx)−nsin(x) as x approaches 0.

Note that Xcas does not complain about a possibility of n being equal to 1.

Find the limit of √x+√x+√x−√x as x approaches +∞.

Find the limit of √1+x+x²/2− e^x/2/(1−cos(x))sin(x) as x approaches 0.