Calculus S. Awad, Ph.D. M. Corless, M.S.E.E. D. Cinpinski E.C.E. Department University of Michigan-Dearborn Math Review with Matlab: Limits

Calculus:Limits 2 n Why Use Limits? Why Use Limits? n Determinate and Indeterminate Functions Determinate and Indeterminate Functions n General Limits General Limits n Limit Command Limit Command n Indeterminate Example Indeterminate Example n Determinate Example Determinate Example n Discontinuous Example Discontinuous Example n Restricted Constant Example Restricted Constant Example

Calculus:Limits 3 Why Use Limits? n Many functions are not defined at certain points n However values of the function arbitrarily close to these points may be well defined n Limits are often used to determine the value of a function around (close to) an undefined point n Limits can also be used to approximate values around points of discontinuity in a function

Calculus:Limits 4 Determinate and Indeterminate n A limit is said to be Determinate if the function is defined at the limit point The limit is still considered determinate if the result f(a)=  n A limit is said to be Indeterminate if the function is NOT defined at the limit point n Two common examples of indeterminate limits include:

Calculus:Limits 5 General Limits n In general, the limit at an indeterminate limit point can be evaluated as: Where  is an infinitesimally small number and: As a   becomes arbitrarily close to a, then f(a  ) becomes arbitrarily close to the limit l

Calculus:Limits 6 Right and Left Limits Consider  is an infinitesimally small positive number The Limit from the Right is defined as: The Limit from the Left is defined as: n If the terms right or left are not specified when evaluating a limit, it is assumed that: n If they are not equal then the right or left limit must be specified before evaluating

Calculus:Limits 7 Limit Command limit(f,x,a) takes the limit of the symbolic expression f as x approaches a limit(f,a) uses findsym(f) as the independent variable and takes the limit as it approaches a limit(f) uses findsym(f) as the independent variable and takes the limit as it approaches 0 limit(f,x,a,'left') takes the limit from the left limit(f,x,a,'right') takes the limit from the right

Calculus:Limits 8 Indeterminate Example n Analyze the function: n f(x) is obtained by dividing the two continuously defined functions plotted to the right n However f(x) is undefined at x = 0 since:

Calculus:Limits 9 Defined Limit Around 0 n Even though f(x) is undefined at x=0 n f(x) is defined as x approaches 0 from the left n f(x) is defined as x approaches 0 from the right Undefined Defined

Calculus:Limits 10 Evaluate Limit Using Delta n The limit can be determined by setting delta to a small value and interpreting the results » delta=sym('1/1000'); » lim_left =eval(sym('sin(-delta)/(-delta)')); » lim_right=eval(sym('sin(delta)/delta')); » vpa(lim_left,10) ans = » vpa(lim_right,10) ans =

Calculus:Limits 11 Verify Using Limit Command Use the limit command to verify the previous results » f_zero=sin(0)/0 f_zero = NaN n The limit from the left is equal to the limit from the right Indeterminate » syms x » f=sin(x)/x; » limit_left=limit(f,x,0,'left') limit_left = 1 » limit_right=limit(f,x,0,'right') limit_right = 1 » limit=limit(f) limit = 1

Calculus:Limits 12 Plot of sin(x) / x The ezplot command can be used to plot the function and graphically verify the result » ezplot(f,[-15,15]) » grid on

Calculus:Limits 13 Determinate Limit Example n Given the determinate symbolic function f: n Use Matlab to find the limit of f as x approaches 0 n Use Matlab to find the limit of f as x approaches a

Calculus:Limits 14 » syms a x b » f=sin(a*x+b); Limit as x approaches 0 n Since x is the Matlab default independent variable and the default limit point is 0 the limit can simply be found using: » lim1 = limit(f) lim2 = sin(b)

Calculus:Limits 15 Limit as x approaches a n Since x is the Matlab default independent variable and the default limit point is 0 the limit can simply be found using: » lim2=limit(f1,a) lim2 = sin(a^2+b) » lim2=limit(f1,x,a) lim2 = sin(a^2+b) n Of course the independent variable can be explicitly specified to return the same results

Calculus:Limits 16 Discontinuous Example n Given the function f(x): n Graphically verify the results by using Matlab to plot f(x) over the region of interest n Use Matlab to determine the following limits of f as x approaches -1, 0, and +1

Calculus:Limits 17 Limits at Defined Points n Use Matlab to evaluate the limits at x=-1 and x=0 » syms f x » f=1/(x-1); » f_neg1=limit(f,x,-1) f_neg1 = -1/2 » f_zero=limit(f,x,0) f_zero = n Matlab returns a numerical result for each, implying that the limits from the right and left are the same

Calculus:Limits 18 Limit at Discontinuity n Finding the limit as x approaches 1 returns NaN (Not a Number) » f_pos1=limit(f,x,1) f_pos1 = NaN Discontinuity at x=1 n This implies that the function is discontinuous at x=1 and that the limit from the left does not equal the limit from the right

Calculus:Limits 19 Left and Right Limits n Must explicitly find the limit from the left and from the right » f_pos1_lft=limit(f,x,1,'left') f_pos1_lft = -inf » f_pos1_rght=limit(f,x,1,'right') f_pos1_rght = inf

Calculus:Limits 20 Verify Results n Use Matlab to plot f(x) and verify the previous results » ezplot(f) » axis([ ]) » grid on

Calculus:Limits 21 Restriction on Constants n Sometimes it is useful to put restrictions on symbolic constants when evaluating the limits on symbolic expressions x=sym('x','real') restricts x to be real x=sym('x','positive') restricts x to be real and positive x=sym('x','unreal') puts no restrictions on x and can be used to undo any previous restrictions

Calculus:Limits 22 Restricted Constant Example n Given the continuous function of: n Determine the limit of f(x) as x approaches infinity for the following cases: u Where k 2 is any real number u Where k 2 is a positive real number n We will not discuss the case where k 2 is considered complex

Calculus:Limits 23 Limit for Real k 2 n If k 2 is assumed to be any real number, the limit is dependent upon the sign of k 2 » syms k1 k2 x » f=k1*atan(k2*x); » k2=sym('k2','real'); » lim_k2real = limit(f,x,inf) lim_k2real = 1/2*signum(k2)*pi*k1

Calculus:Limits 24 Limit for Positive k 2 » k2=sym('k2','positive'); » lim_k2pos = limit(f,x,inf) lim_k2pos = 1/2*pi*k1 n The limit as x goes to infinity when k 2 is known to be a positive real number returns a simpler result n The result is consistent with the more general result n This may be useful when simplifying symbolic expressions

Calculus:Limits 25 Summary n General use of limits n Determinate and indeterminate functions n Limits from the right and left n Evaluating limits of symbolic expressions n Restrictions on symbolic constants