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# 18. Limits

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## 18.1 Functions and Variables for Limits

Option variable: lhospitallim

  Default: 4

  `lhospitallim` is the maximum number of times L'Hospital's rule is used in `limit`. This prevents infinite looping in cases like `limit (cot(x)/csc(x), x, 0)`.

  Categories: Limits

Function: limit (expr, x, val, dir)
Function: limit (expr, x, val)
Function: limit (expr)

  Computes the limit of expr as the real variable x approaches the value val from the direction dir. dir may have the value `plus` for a limit from above, `minus` for a limit from below, or may be omitted (implying a two-sided limit is to be computed).

  `limit` uses the following special symbols: `inf` (positive infinity) and `minf` (negative infinity). On output it may also use `und` (undefined), `ind` (indefinite but bounded) and `infinity` (complex infinity).

  `lhospitallim` is the maximum number of times L'Hospital's rule is used in `limit`. This prevents infinite looping in cases like `limit (cot(x)/csc(x), x, 0)`.

  `tlimswitch` when true will allow the `limit` command to use Taylor series expansion when necessary.

  `limsubst` prevents `limit` from attempting substitutions on unknown forms. This is to avoid bugs like `limit (f(n)/f(n+1), n, inf)` giving 1. Setting `limsubst` to `true` will allow such substitutions.

  `limit` with one argument is often called upon to simplify constant expressions, for example, `limit (inf-1)`.

  `example (limit)` displays some examples.

  For the method see Wang, P., "Evaluation of Definite Integrals by Symbolic Manipulation", Ph.D. thesis, MAC TR-92, October 1971.

  Categories: Limits

Option variable: limsubst

  default value: `false` - prevents `limit` from attempting substitutions on unknown forms. This is to avoid bugs like `limit (f(n)/f(n+1), n, inf)` giving 1. Setting `limsubst` to `true` will allow such substitutions.

  Categories: Limits

Function: tlimit (expr, x, val, dir)
Function: tlimit (expr, x, val)
Function: tlimit (expr)

  Take the limit of the Taylor series expansion of `expr` in `x` at `val` from direction `dir`.

  Categories: Limits

Option variable: tlimswitch

  Default value: `true`

  When `tlimswitch` is `true`, the `limit` command will use a Taylor series expansion if the limit of the input expression cannot be computed directly. This allows evaluation of limits such as `limit(x/(x-1)-1/log(x),x,1,plus)`. When `tlimswitch` is `false` and the limit of input expression cannot be computed directly, `limit` will return an unevaluated limit expression.

  Categories: Limits

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