Use the Calculus > Limit command or press Ctrl+Shift+L to find the limit of an expression as one of its variables approaches a point.  This command allows you to select the limit variable, the limit point, and the direction from which the point is approached on the real line.


Alternatively, the limit of the expression u as the variable x approaches the point a can be entered by typing in an expression of the form LIM(u, x, a).  a defaults to 0.  For example, to use the limit to determine the derivative of x², simplify the expression

LIM(((x+h)^2-x^2)/((x+h)-x), h, 0)


If there is a discontinuity in u at x=a, the limit as x approaches a from the right may differ from the limit as x approaches a from the left.  The right-hand limit can be entered by calling LIM with a positive number, such as 1, as its fourth argument.  For example, to determine the limit of SIGN(x) as x approaches 0 from the right, simplify the expression

LIM(SIGN(x), x, 0, 1)


The left-hand limit can be entered by calling LIM with a negative number, such as -1, as its fourth argument. For example, to determine the limit of SIGN(x) as x approaches 0 from the left, simplify the expression

LIM(SIGN(x), x, 0, -1)


Note that the two-sided limit

LIM(SIGN(x), x, 0)

simplifies to ±1.


As another example, the right-hand limit

LIM(1/x, x, 0, 1)

simplifies to ∞ (plus infinity), the left-hand limit

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

simplifies to -∞ (minus infinity), and the two-sided limit

LIM(1/x, x, 0)

simplifies to ±∞ (plus or minus infinity).


Limits as a variable increases without bounds can be found by calling LIM with infinity as the limit point.  Infinity can be entered by clicking on the ∞ on the math symbol toolbar, pressing Ctrl+0, or by typing inf.  For example, to determine the limit of a·x/(x+1) as x approaches plus infinity, simplify the expression

LIM(a·x/(x+1), x, inf)


Both LIM and SUBST (see the Simplify > Variable Substitution commandGP2.L1) can be used to substitute a value for a variable in an expression.  However, the limit of an expression may simplify to a unique value; whereas substitution may simplify to an ambiguous answer such as ±inf, ±1, or ? (unknown).  For example,

LIM(x/SIN(x), x, 0)

simplifies to 1; whereas

SUBST(x/SIN(x), x, 0)

simplifies to ?. 


To facilitate using LIM for substitution, its second argument can be a vector of variables, instead of just a variable.  Then its third argument should be a vector of values for those variables.  For example,

LIM(x^2+y^2, [x,y], [2,3])

simplifies to 13 (i.e. 2²+3²).



You can use LIM to substitute values for variables in function definitions (see the Author > Function Definition command1ER8JVI).  For example, to define the function SLOPE which returns the slope of the secant line through the points u(a) and u(b), enter the expression

SLOPE(u,x,a,b) := (LIM(u,x,b)-LIM(u,x,a))/(b-a)


Then, to determine the slope of the secant line to the curve EXP(x) through the points x=1 and x=2, simplify the expression

SLOPE(EXP(x),x,1,2)


Other Calculus commandsCalculus_commands 

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