# limits - 2A: Lecture 3   Section 2.2 The limit of a...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2A: Lecture 3   Section 2.2 The limit of a function     Group activity: Suppose f is a function defined  around a point a.     We say that   lim f(x)=L if… x a   2  Informal definition:     We say that the limit of f(x) as x approaches a   is equal L and we write                                   if we can make f(x) as close to L as we want   by taking x sufficiently close to a, but not equal to a.      3        lim f(x) = 3 x -2 TRUE FALSE   4    lim f(x) = 3 x -2 TRUE FALSE If x approaches -2 (from either direction), f(x) approaches 3. We can make f(x) as close to 3 as we want, by taking x sufficiently close to -2 (but not equal to -2). So lim f(x)=3. x   -2 5  Note that f(-2)=2, not 3, but this does not matter at all. lim f(x)= - 4 x 3 TRUE FALSE   6      lim f(x)= - 4 x TRUE FALSE 3 If x approaches 3 from the left, f(x) approaches 0. If x approaches 3 from The right, f(x) approaches -4. So lim f(x). x   3 7  does not exist.      E             Choose values of x that get closer and  closer to x=2 (but are not =2) and plug  these values into the function.    x >2 2 .5 2 .1 2 .0 1 2.001 2.0001 2.00001   f(x ) 3 .4 3.857142857 3.985074627 3.998500750 3.999850007 3.999985000 x <2 1 .5 1 .9 1 .9 9 1.999 1.9999 1.99999 f(x ) 5 .0 4.157894737 4.015075377 4.001500750 4.000150008 4.000015000 Take a guess!    The limit of f(x) as x approaches 2 is …           8           x >2 2 .5 2 .1 2 .0 1 2.001 2.0001 2.00001   f(x ) 3 .4 3.857142857 3.985074627 3.998500750 3.999850007 3.999985000 x <2 1 .5 1 .9 1 .9 9 1.999 1.9999 1.99999 f(x ) 5 .0 4.157894737 4.015075377 4.001500750 4.000150008 4.000015000   The values of f approach 4.     We guess that                        OK!                                                                         9                   Choose values of x that get closer and   closer to x=2 (but are not =2) and plug  these values into f.     same table of values as before!     GUESS:                 Limits are concerned with what goes on  around the point (not at the point itself!)    10     f is not defined at o !  Choose values of x that get closer and  closer to x=0 (but are not =0) and plug  them into the function.  >0 1 0 .1 0 .0 1 0.001 0.45969769 0.04995835 0.00499996 0.00049999 <0 -1 -0.45969769 - 0 .1 -0.04995835 - 0 .0 1 -0.00499996 -0.001 -0.00049999   The values of f approach 0.     We guess that        OK      11  Can we always guess the limit  this way?     12               f is not defined at o !  Choose values of x that get closer and  closer to x=0 (but are not =0) and plug  these values into f.   t 1 0 .1 0 .0 1 0.001 f(t) -1 1 1 1 t -1 - 0 .1 - 0 .0 1 -0.001 f(t) -1 1 1 1       Guess: the limit is 1.           13    This guess is quite WRONG!!!        f oscillates rapidly around 0, and never  settles down to a fixed value.     The limit of f as x approaches 0 does not  exists!                14        (Heaviside or step function)     The function doesn’t settle down to a  single number as we move in towards 0.    If we approach 0 from the right, the  values of f move toward 1.  If we approach 0 from the left, the values  of f move toward 0.    The limit of f as x approaches 0 DNE.   15  Limit a  We say that the limit of f as x approaches a is equal to  L, and we write                                   if we can make f(x) as close to L as we want   by taking x sufficiently close to a but different from a.   Right-handed limit a We say that the limit of f as x approaches a from the  right is equal to L and we write                                    if we can make f(x) as close to L as we want   by taking x sufficiently close to a but bigger than a.   Left-handed limit a We say that the limit of f as x approaches a from the  left say that the limd wfe ws xe  pproaches a from the  We  is equal to L an it o  f a rit  a   eft is equal to L and we write  l                                     if we can make f(x) as close to L as we want                                     by te cian makff ifcients cloose toLa but e wanlte r than a. if w ak ng x sue (x) a ly cl se to     as w smal   by taking x sufficiently close to a but smaller th6an a.   1       f is not defined at ‐4.   The limit exists:       f is defined at +1.   The limit does not exist:                            f is defined at +6.    The limit exists:        17  f(2) lim f(x) x2 + f(3) lim f(x) x3 + f(1) lim f(x) x1 + lim f(x) x2 x2 - lim f(x) x3 x3 - lim f(x) x1 x1 - lim f(x) lim f(x) lim f(x)   18  f(2) 4 lim f(x) 4 x2 + f(3) ~2.5 lim f(x) 1 x3 + f(1) ~1.5 lim f(x) 2 x1 + lim f(x) 4 x2 x2 - lim f(x) ~2.5 x3 x3 - lim f(x) 2 x1 x1 - lim f(x) 4 lim f(x) DNE lim f(x) 2   20    Choose values of x that get closer  and closer to x=0 (but are not =0)  and plug these values into f.  x - 0 .1 - 0 .0 1 -0.001 -0.0001   1/x -1 0 -1 0 0 -1 0 0 0 -10000 x 0 .1 0 .0 1 0.001 0.0001 1/x 10 100 1000 10000   As we make x smaller and smaller,  f(x)= 1/x gets larger and larger   but it retains the same sign as x     Guess: The limit does not exist.      23    We say that x=0 is a vertical  asymptote for the graph.              24          Again the limit is infinite.    We say that x=0 is a vertical asymptote   for the graph.    25            Infinite Limit We say that the limit of f as x approaches a is + infinity, and we write if we can make f(x) arbitrarily large, by taking x sufficiently close to a, but different from a. We say that the limit of f as x approaches a is -infinity, and we write if we can make f(x) arbitrarily large, by taking x sufficiently close to a, but different from a.     The definition of right and left infinite limits is similar.    26  Vertical Asymptote The function f(x) has a vertical asymptote at in any of the following cases:                                           27  x=‐2 is a vertical  asymptote.          x=+3 is a vertical  asymptote.           x=0 is a vertical  asymptote   28     The tangent function has many vertical  asymptotes...            29  3 WAYS IN WHICH THE LIMIT MAY FAIL TO EXIST     The limit from the left is different from the limit from the right.    a           The limit (from the  right or from the left) does not exist  because the function oscillates.      The limit is infinite.    30  ...
View Full Document

## This note was uploaded on 12/07/2010 for the course MATH MATH 2A taught by Professor Alessandrapantano during the Fall '10 term at UC Irvine.

Ask a homework question - tutors are online