( ) describes how the values of f ( x) behave as x approaches a and not at a,

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1 Math 1205 Calculus Sec. 2.2: Intuitive Introduction To Limits Abbreviations: wrt = with respect to! = for all! = there exists!= therefore Def n = definition Th m = Theorem sol n = solution! = perpendicular iff or! = if and only if pt = point fn = function eq = equation! =is an element of st = such that I. Introduction A. What does the word it mean? Webster s Dictionary gives a few different meanings to this word: (1) a boundary or boundary line, (2) that which terminates, circumscribes, restrains or confines; the utmost extent; a point not to be passed, (3)Exchanges: a maximum or minimum price at which a broker is to buy or sell,(4) Games: In betting, the sum agreed on as the greatest by which stakes may be increased at one time,(5) Hunting: the quantity of game or fish that may be taken legally in a specified period, (6) Logic: a determining feature. The dictionary also gives math definitions as well. B. What is a speed it? II. Limit A. Informal Definition of Limit 1. Def n : Let f (x) be a function. Suppose, as x approaches a (but x! a ), that the corresponding values f (x) approach a number L. Then we say that L is the it L of f (x) as x approaches a and we write: f ( x) = L ( the it of x!a f(x), as x approaches a, equals L ). 2. ( ) describes how the values of f ( x) behave as x approaches a and not at a, itself or that f ( x) approaches the it value L as x approaches a. 3. x! a means that x approaches from both sides of a. B. The Value of f (a) is Not Relevant to ( ) 1. The specific value f (a) has no bearing on the value of 2. f (a) does not have to be defined. C. Tabular and Graphical Approaches EXAMPLES: 1. Evaluate x!1 ( x 3 + 4x), i.e., how does the graph of f(x)=x 3 +4x behave near x=1? x! 1 " (from left of 1) f (x) = x 3 + 4x x! 1 + (from rt of 1) f (x) = x 3 + 4x Note: (1) x! a + means that x approaches a from the right or larger values. (2) x! a " means that x approaches a from the left or smaller values. ( )

2 GRAPHICAL APPROACH: From the graph we can see that as x approaches 1, the function approaches Therefore, either method indicates that ( ) = x 3 + 4x x!1 2. Evaluate # % x 2 " 4& x!"2 $ x + 2 '. Notice that f(-2) is not defined; x values near -2 f ( x) = x 2! 4 x Indeterminate GRAPHICAL APPROACH: From the graph or the table to the left, it is clear that f(x) gets "close" to the value 4 as x approaches -2, either from the left or from the right. What we say is that f(x) approaches the it -4 as x approaches -2. We write: # % x 2 " 4& = x!"2$ x + 2 ' Note: You can reduce the fn f ( x) = x 2! 4 x + 2 to f ( x) = x! 2, x! "2.

3 4. Evaluate f ( x) if f ( x) = x!4 #% x 2! 5x + 4 ; x " 4 $ x! 4 & % x - 5 ; x = 4 GRAPHICAL: x values near 4 f(x) Using either method, we see that the f ( x)= x!4 REMEMBER: The existence of a it does not depend on how the function may or may not be defined at x=a. x "sin( x) 5. Evaluate x 3 x! 0 x! sin( x) x GRAPHICAL: x "sin( x) = x 3 But if we use values even closer to zero, the table values below are computed. x! 0 x! sin( x) x GRAPHICAL: The table values look like the answer should be zero, the error arises due to the calculator giving false values since x and sin(x) differ in decimal places further out than your calculator uses. An error in the graph will also occur with this extremely small domain for x.! x "0 ( ) x 3 = 1 6. x #sin x

4 III. One-Sided Limits A. Definitions 1. Left-hand Limit / Limit from Below: (x approaches a from smaller values) Def n : Let f (x) be a function. Suppose, as x approaches a from the left, that the ( ) = L. corresponding values f (x) approach a number L. Then we say that L is the lefthand it L of f (x) as x approaches a and we write: " 2. Right-hand Limit / Limit from Above: (x approaches a from larger valurs) Def n : Let f (x) be a function. Suppose, as x approaches a from the right, that the corresponding values f (x) approach a number L. Then we say that L is the right-hand it L of f (x) as x approaches a and we write: f ( x) = L. + B. Th m : f ( x)=l iff f ( x)= x!a x!a - x!a + ( )=L, where a, L are real numbers. C. EXAMPLES: Evaluate the following its. 1. Given g x " ( ) = 5x! 5 ; x < 3 # $ x + 3 ; x > 3 a. g( x) = x!3 " b. g( x) = x!3 + c. g( x)= x! 3 2. ( 5 " x) = x!5 " 3. ( 5 " x) = x! " x x!5 ( ) = 5. x x

5 6. Evaluate the following its using the graph below. x!"2 ( ) = f -2 ( ) = f ( -1 ) = x!"1 " ( ) = x!"1 + ( ) = x!"1 ( ) = x!0 " ( ) = x!0 + x!1 ( ) = f ( 1) = ( ) = x!0 ( ) = IV. Why Limits Fail to Exist A. Both the right-had and left-hand its exist, but they are not equal, i.e., f ( x) # f ( x ). (jump in fn values at x=a.) " + EXAMPLE: f (x) = x x f( x) does not exist because f ( x) # f ( x) " x! 0 +

6 B. One or both of the right-hand and left-hand its fail to exist. 1. The function values, f (x), become unbounded as x approaches a, i.e., ( )! " or EXAMPLE: f (x) = 1 x 2 ( )! #" (function grows without bound) f( x) does not exist because f( x)! " 2. The function values, f (x), does not oscillate down to a specific number EXAMPLE: f (x) = sin( 1 x ) sin 1 x ( ) does not exist due to the function not oscillating down to a specific value. x! 0 " sin( 1 x ) x! 0 + sin( 1 x ) Function is not defined from one side of a. x does not exist because you cannot approach x=0 from the left. NOTE: only x!0 + x exists and =0