Univariate function limits

### 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 limx→ 0−1/x, input:

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

or:

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

−∞ |  |  |  |  |  |  |  |  |  |

To find limx→ 0+1/x, input:

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

or:

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

+∞ |  |  |  |  |  |  |  |  |  |

To find limx→ 01/x, input:

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

or:

limit(1/x,x,0)

or:

limit(1/x,x=0)

∞ |  |  |  |  |  |  |  |  |  |

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.

limit((n*tan(x)-tan(n*x))/(sin(n*x)-n*sin(x)),x=0)

2 |  |  |  |  |  |  |  |  |  |

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 +∞.

limit(sqrt(x+sqrt(x+sqrt(x)))-sqrt(x),x=+infinity)

1
2

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

limit((sqrt(1+x+x^2/2)-exp(x/2))/((1-cos(x))*sin(x)),x,0)

− |
1
6