Limits

Problem: determine the "behaviour" of the funtion as x gets "close" to zero.

Approach 1: Make a table, or tables, of values (i.e. by "plugging in").

Approach 2: Look at a graph of y = .

Approach 1: Plugging in:

The table suggests that as x "approaches" zero, "approaches" one. In mathematical symbols, we would write that and say out loud: " The limit as x approaches zero of is one."

Approach 2: Looking at a plot.

> plot(sin(x)/x,x,colour=black);

By looking at the graph, it appears that the closer x is to 0 (look at the x-axis), the closer is to 1 (look at the y-axis) and again we would guess that .

Informal Definition.

Definition : (Stewart, page 91) : "We write and say " the limit of f(x), as x approaches a, equals L" if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a but not equal to a ."

i.e. means that as x gets closer and closer to a, f(x) gets closer and closer to L.

Very easy examples :

, , .

A Slightly Harder Example.

Find by making a table of values.

The table above makes it look like that, as x approaches -2 from the left (i.e. through values smaller than -1) , f(x) approaches -5 and, to indicate the direction of approach we would write .

>

The table above makes it look like that, as x approaches -2 from the right (i.e. through values larger than -2) , f(x) approaches -5 and, to indicate the direction of approach we would write .

Since both approaches give the same answer (assuming our "guesses" are correct) we would say that the limit exists and write .

One Sided Limits.

Definition : (see Stewart, page 95) is the "left hand limit of f as x approaches a from the left." i.e. means that as x approaches x through numbers smaller (to the left of) than a, f(x) approaches L.

There is an symmetric definition for the "right hand limit:" . (p. 95)

Revisit a "Slightly Harder Example".

> expr:=(2*x^2+3*x-2)/(x+2);

>

> factor(2*x^2+3*x-2);

> simplify(expr);

Now it is easy to see that = = -5

Important question : Why is it true that is not equal to but their limits are equal????????

Sometimes Does Not Exist (DNE)

Make a table:

The table shows that as x -> 0 from the right , 1/x becomes BIG without bound. The limit DNE but to describe the behaviour in the table we write .

The table also shows that as x -> 0 from the left , 1/x becomes BIG NEGATIVE without bound. The limit DNE but to describe the behaviour in the table we write .

Look at the corresponding behaviour in the graph of y = 1/x.

> plot(1/x,x=-4..4,y=-10..10,discont=true,colour=black);

> g:=sin(1/x):

> plot(g,x,colour=black);

> plot(g,x=-1..1,colour=black);

> plot(g,x=-1/10..1/10,colour=black);

Certainly the plots show Bad Behaviour close to x = 0 -- and the closer you get, the "badder" the behaviour. It sure doesn't look like is approaching any particular number as x -> 0 and we should write DNE. Try to analyze the behaviour of as x -> 0 more mathematically by solving the equation or -1.

You Can't Always Trust a Table!

> h:=(sqrt(x^2+9)-3)/(x^2);

The table shows some odd behaviour as x->0 but eventually it settles down and appears that

Let's look at the graph:

> plot(h,x,colour=black);

Certainly it looks like the limit exists and is close to 0.16.

Let's ask Maple:

> limit((sqrt(x^2+9)-3)/(x^2),x=0);

Now prove it!!!!!!!!

Infinite Limits

e.g.

> k:=(x^2+1)/(x^2-x);plot(k,x=-4..4,y=-12..12,discont=true,colour=black);

Clearly, there are problems at x = 0 and x = 1. Let's ask Maple:

> limit(k,x=0);limit(k,x=0,left);limit(k,x=0,right);

> limit(k,x=1);limit(k,x=1,left);limit(k,x=1,right);

Look at the corresponding behavior in the graph.

Look at the blackboard for the analysis!

Vertical Asymptotes or "Infinite Limits"

Definition : "Infnite Limits": means that the values of f(x) grow without bound as x approaches a. Similar statements can be written for , , , etc...

Note: Infinite limits are not "real" limits but are only a shorthand method for describing the behaviour of a function close to certain "bad spots."

definition : (see page 58) The line x = a is a vertical asymptote if there is an infinite limit as x --> a (from any or all directions)

Note: For all Rational Functions there is always a VA at x = a if but . See the above example.

The Limit Laws & Evaluating Limits

If and then:

(See tables on page 61& 62 in Stewart)

1) (The limit of a sum is the sum of the limits.)

3) ; where c is a constant. (The limit of a constant times a function is the constant times the limit of the function.)

4) (The limit of a product is the product of the functions.)

5) if (The limit of a quotient is the quotient of the limits if the limit of the denominator is not zero.)

6) (The limit of a power is the power of the limit.)

7) (The limit of a constant is the constant.)

8) (As x gets close to a, x gets close to a!!!!)

9) (Combine #6 and #8 or #4 and #8.)

11) (The limit of a root is the root of the limit. If n is even then we assume that . )

Consequences of the Limit Laws

Show that if then .

Generalization: If f is any polynomial then . (Polynomials have the "plug in property" with respect to limits.)

Generalization: If f and g are polynomials and and then = = = R( a ). (On their domains, rational functions have the "plug in property" with respect to limits.)

Strategies for evaluating limits

Limits are evaluated by applying the limit laws -- and often this amounts to simply "plugging in" . A problem only develops if in trying to apply the laws an undefined expression (such as ) is produced. If you produce an expression of the form then the limit DNE. At that point you must analyze the function to see if there is a vertical asymptote (infinite limit) at that x value.

If applying the limit laws produces an expression of indeterminate form such as then you must do more work! Indeterminate forms contain no information (i.e. you cannot say from 0/0 whether or not the expression has a limit as x approaches a). In such cases, you must try manipulating the expression, using only legal operations, until you produce a form to which the limit laws can be applied fruitfully. Comon manipulations include: simplifying and/or factoring and canceling, rationalizing numerators or denominators, simplifying any complex fractions, anything else that is legal and works!

** Indeterminate: , , , (These all mean more work when they arise in a limit calculation.)

** Determinate: if , if , , , (These all yield information (i.e. a meaningful answer) when they arise in a limit calculation.)

Note: one-sided limits are normally only evaluated separately at numbers where there is a VA or at numbers where the function changes definition (such as might occur in a piecewise defined function ).

e.g. Prove the following Maple conclusions.

> Limit((x^2+2*x-15)/(x+5),x=-5)=limit((x^2+2*x-15)/(x+5),x=-5);

> Limit((sqrt(x^2+9)-3)/x^2,x=0)=limit((sqrt(x^2+9)-3)/x^2,x=0);

> Limit(1/(t*sqrt(1+t))-1/t,t=0)=limit(1/(t*sqrt(1+t))-1/t,t=0);

>

> Limit(abs(3-x)/(x-3),x=3)=limit(abs(3-x)/(x-3),x=3);

Note: in this context, undefined means DNE.

> f:=piecewise(x<=0,x+1,x>0,2-x^2);

> Limit(f,x=-1)=limit(f,x=-1);

> Limit(f,x=0,right)=limit(f,x=0,right);

> Limit(f,x=0,left)=limit(f,x=0,left);

> Limit(f,x=0)=limit(f,x=0);

> plot(f,x=-4..4,y=-4..4,discont=true,colour=black);

The Squeeze Theorem

The squeeze theorem is one of the most aptly named theorems in mathematics. Informally, it says that if you are interested in (and you are aren't you?) and you happen to know that g(x) is squeezed between two other functions, both of which have the same limit at a then g must have the same limit!

Remember the e.g. of ? We concluded that did not exist. Now let's consider the function . Check out the following graph of g which includes the lines y = x and y = -x.

> plot({x*sin(1/x),abs(x),-abs(x)},x=-0.1..0.1,y=-0.1..0.1,colour=black,numpoints=2000);

You should be able to see that g is squeezed between the functions y = x and y= -x for all x's close to zero. It follows, by the squeeze theorem, that since = 0 = .

Squeeze Theorem : (see page 67 in Stewart ) If for all x in an open interval that contains a (except possibly at a ) and = L = then = L.

e.g. Show that = 0.

> plot({sin(x)/x,1/x,-1/x},x=-30..30,y=-1..1,colour=black);