Symbolic Math Toolbox

Limits

The fundamental idea in calculus is to make calculations on functions as a variable "gets close to" or approaches a certain value. Recall that the definition of the derivative is given by a limit

provided this limit exists. The Symbolic Math Toolbox enables you to calculate the limits of functions directly. The commands

• syms h n x
  limit( (cos(x+h) - cos(x))/h,h,0 )
  

which return

• ans =
  -sin(x)
  

and

• limit( (1 + x/n)^n,n,inf )
  

which returns

• ans =
  exp(x)
  

illustrate two of the most important limits in mathematics: the derivative (in this case of cos x) and the exponential function.

One-Sided Limits

You can also calculate one-sided limits with the Symbolic Math Toolbox. For example, you can calculate the limit of x/|x|, whose graph is shown in the following figure, as x approaches 0 from the left or from the right.

To calculate the limit as x approaches 0 from the left,

enter

• limit(x/abs(x),x,0,'left')
  

This returns

• ans =
   -1
  

To calculate the limit as x approaches 0 from the right,

enter

• limit(x/abs(x),x,0,'right')
  

This returns

• ans =
  1
  

Since the limit from the left does not equal the limit from the right, the two- sided limit does not exist. In the case of undefined limits, MATLAB returns NaN (not a number). For example,

• limit(x/abs(x),x,0)
  

returns

• ans =
  NaN
  

Observe that the default case, limit(f) is the same as limit(f,x,0). Explore the options for the limit command in this table, where f is a function of the symbolic object x.

Mathematical Operation	MATLAB Command
	limit(f)
	limit(f,x,a) or limit(f,a)
	limit(f,x,a,'left')
	limit(f,x,a,'right')

Calculus

Integration