Consider the function Note that for 1 from the right from the left LIMIT OF A FUNCTION Consider the function Note that for Investigate the behavior of f(x) as x goes very very close to 1. from the left 1 from the right Take x close to 1 but less than 1 Take x close to 1 but greater than 1

Let by assigning values of x very very close to 1 but less than 1. 0.5 0.75 0.9 0.99 0.999 0.9999 f(x) 5 6 6.5 6.8 6.98 6.998 6.9998 In symbol, we write this as ONE-SIDED LIMIT : LEFT HAND SIDE LIMIT

Let by assigning values of x very very close to 1 but greater than 1. 2 1.5 1.25 1.1 1.01 1.001 1.0001 f(x) 9 8 7.5 7.2 7.02 7.002 7.0002 In symbol, we write this as ONE-SIDED LIMIT : RIGHT HAND SIDE LIMIT

If one-sided limits both exist and are equal TWO-SIDED LIMIT If one-sided limits both exist and are equal to one another then two-sided limit exists and is equal to the same value. Note that function value at x = 1 does not exists but limit of the function as x approaches close 1 exists and is equal to 7. So limit of a function is that number to which the function value is going very near to as we take x very close to 1.

We express this by saying that the limit of f(x) 7 x approaches 1 from the left x approaches 1 from the right 6 8 Observe from both tables that as x gets closer to 1 (on either side of 1), f(x) gets closer to 7. We express this by saying that the limit of f(x) as x approaches 1 is equal to 7. We write this in symbol as This leads to the intuitive definition of limit.

INTUITIVE DEFINITION OF LIMIT Let f(x) be defined on an open interval containing a, except possibly at a itself. We say that the limit of the function f(x) as x approaches a is equal to L , written as if f(x) gets closer and closer to L, whenever x gets closer and closer to a (on either side of a). Going back to Tables 1 and 2, we see that the closer x is to 1, the closer f(x) is to 7. Also, we can make the value of f(x) as close to 7 as we want by taking x sufficiently close to 1.

Using inequalities, Using inequalities, For instance, In Table 1, if x = 0.9 , f(x) =6.8 In Table 2 , if x = 1.1 , f(x) = 7.2 That is , if x is within a distance 0.1 from 1 , f(x) is within a distance 0.2 from 7 Using inequalities, ( 1 ) In Table 1, if x = 0.99 , f(x) = 6.98 In Table 2 , if x =1 .01 , f(x) = 7.02 That is , if x is within a distance 0.01 from 1, f(x) is within a distance 0.02 from 7 ( 2 ) Using inequalities,

This shows that we can make the absolute value of the difference For instance, In Table 1, if x = 0.999 , f(x) =6.998 In Table 2 , if x = 1.001 , f(x) = 7.002 That is , if x is within a distance 0.001 from 1 , f(x) is within a distance 0.002 from 7 Using inequalities, ( 3 ) This shows that we can make the absolute value of the difference between f(x) and 7 as small as we want by making the absolute value of the difference between x and 1 small enough. This situation can be made formally by using the Greek letters (epsilon) and (delta) for these small differences.

Hence, we say that for any given positive number , there is a positive number such that ( 4 ) In (1) In (2) In (3) Note that is chosen first and the value of depends on . Here, and (4) is equivalent to

Graphically, this means that if x lies in the open interval , then f(x) lies in the open interval .

Thus, we have the following formal definition of limit Or equivalently, For every there exists

LIMIT RULES The formal definition of limit does not give us a way to evaluate the limit of a function but it enables us to verify that the limit is correct. To evaluate limits, we have the following limit rules whose proofs are based on the formal definition of limit.

Rule 1 : If f(x) = x ( the identity function ) , then for any a Rule 2 : If f(x) = k ( the constant function ) , then for any a If and where L and M are real numbers, then Rule 3 : (Sum/difference Rule) Rule 4 : (Product Rule)

Rule 9 : Limit of a Rational Function Rule 5 : ( Constant multiple ) Rule 6 : (Quotient rule) Rule 7 : (Power rule) The above rules give two immediate results for evaluating limit of a polynomial and a rational function. Rule 8 : Limit of a Polynomial If P(x) is a polynomial function, then Rule 9 : Limit of a Rational Function If P(x) and Q(x) are polynomials, then provided Q(a) 0

Examples

ONE-SIDED LIMIT All limit rules for two-sided limit apply for one-sided limit. (right hand side limit only)

equal Graphically, it means as the graph of f(x) . equal Graphically, it means as the graph of f(x) approaches the same point ( 2, 0 ) as when

( 2, 0 )

Not equal

Graphically, it means as the graph of f(x) does not approach the same point as when .

Not equal Graphically, it means as the graph of f(x) does not . Not equal Graphically, it means as the graph of f(x) does not approach the same point as when

equal Graphically, it means as the graph of f(x) . equal Graphically, it means as the graph of f(x) approaches the same point (3,0) as when

Evaluate the limits and function values if they exist from the graph of f sketched below The domain of f is [0, 5].

Consider the function from the left 2 from the right INFINITE LIMIT Investigate the behavior of f(x) as x goes very very close to 2. from the left 2 from the right Take x close to 2 but less than 2 Take x close to 2 but greater than 2

Let by assigning values of x very very close to 2 but less than 2. 1 1.5 1.75 1.9 1.99 1.999 1.9999 f(x) -1 -2 -4 -10 -100 -1000 -10,000 In symbol, we write this as Note: is not a number. It is a symbol used to indicate that a quantity is decreasing or increasing without bound.

Let by assigning values of x very very close to 2 but greater than 2. 3 2.5 2.25 2.1 2.01 2.001 2.0001 f(x) 1 2 4 10 100 1000 10,000 In symbol, we write this as

vertical asymptote

vertical asymptotes

Theorem : Let n be a positive integer, then Theorem: (Limit of Sum Involving infinity )

Theorem: (Limit of Product Involving infinity )

Theorem: (Limit of Quotient Involving infinity )