Limits MCV4U: Calculus & Vectors What is the value of 1 2 + 1 4 - PDF document

l i m i t s l i m i t s Limits MCV4U: Calculus & Vectors What is the value of 1 2 + 1 4 + 1 8 + 1 16 + . . . ? A visual depiction is below. Limits and Their Properties J. Garvin J. Garvin — Limits and Their Properties Slide 1/23 Slide 2/23 l i m i t s l i m i t s Limits Limits If we use the first four terms of the sequence, then Example 1 2 + 1 4 + 1 8 + 1 16 = 15 16 . Determine the value of lim x → 4 ( x − 3). If we increase the number of terms, we obtain the following: This is a linear function, f ( x ) = x − 3, whose graph is below. Terms 5 6 7 . . . 20 31 63 127 1048575 Sum . . . 32 64 128 1048576 As the number of terms increases, the sum approaches 1. We call this concept a limit . Limits A limit is some value that a function (or sequence) approaches, as the input (or index) approaches some value. J. Garvin — Limits and Their Properties J. Garvin — Limits and Their Properties Slide 3/23 Slide 4/23 l i m i t s l i m i t s Limits Limits The expression lim x → 4 ( x − 3) means “what value does the Example linear function approach as x gets closer to 4?” 1 Determine the value of lim x . By observation, as x → 4, f ( x ) → 1. x →∞ Therefore, we state that lim x → 4 ( x − 3) = 1. This time, we are not approaching a specific value , but ∞ itself. In this example, it is also true that f (4) = 1, but this does Recall that the end behaviour of the function 1 x is defined by not always need to be true. values of x that approach ∞ . Thus, the question can be restated as “does the end behaviour of f ( x ) = 1 x cause it to approach a specific value?” Again, a graph of the function (or a knowledge of its basic properties) is useful. J. Garvin — Limits and Their Properties J. Garvin — Limits and Their Properties Slide 5/23 Slide 6/23

l i m i t s l i m i t s Limits Limits Example x − 3 Determine the value of lim x 2 . x → 0 While it is possible to graph this rational function, an alternative method is to use a table of values that become closer and closer to 0. First, check values that are less than 0. x − 0 . 1 − 0 . 01 − 0 . 001 . . . The graph of f ( x ) = 1 x has a horizontal asymptote at − 3 × 10 6 f ( x ) − 310 − 30100 . . . f ( x ) = 0, and as x → ∞ , f ( x ) → 0. While the function never actually takes on a value of 0, it f ( x ) decreases rapidly, the closer it gets to 0. It appears that 1 gets infinitesimally close to 0 and we say that lim x = 0. as x → 0, f ( x ) → −∞ . x →∞ J. Garvin — Limits and Their Properties J. Garvin — Limits and Their Properties Slide 7/23 Slide 8/23 l i m i t s l i m i t s Limits One-Sided Limits Next, check values that are greater than 0. In the last example, we tested values close to a specific value. A limit that approaches a certain value “from the left” or 0 . 001 0 . 01 0 . 1 x . . . “from the right” is called a one-sided limit . − 3 × 10 6 f ( x ) . . . − 30100 − 310 Left- and Right-Handed Limits For a function f ( x ), we denote as follows: Again, f ( x ) decreases rapidly and as x → 0, f ( x ) → −∞ . • the limit as x approaches a from the left, lim x → a − f ( x ), is Since all values suggest that f ( x ) continues to decrease the called the left-handed limit x − 3 closer it gets to 0, we say that lim = −∞ . • the limit as x approaches a from the right, lim x → a + f ( x ), is x 2 x → 0 Remember that ∞ is not a value. A function can approach called the right-handed limit ∞ , but will never reach it! The values of the left- and right-handed limits may be different, depending on the function, or they may not exist at all. J. Garvin — Limits and Their Properties J. Garvin — Limits and Their Properties Slide 9/23 Slide 10/23 l i m i t s l i m i t s One-Sided Limits One-Sided Limits Example Moving from the left, lim x → 3 − f ( x ) = 2. For the function below, state the values of lim x → 3 − f ( x ) and Moving from the right, lim x → 3 + f ( x ) = 2. x → 3 + f ( x ), if they exist. lim In this case, the left- and right-handed limits are equal. This is not always true. J. Garvin — Limits and Their Properties J. Garvin — Limits and Their Properties Slide 11/23 Slide 12/23

l i m i t s l i m i t s One-Sided Limits One-Sided Limits Example Moving from the right, lim x → 0 + f ( x ) = 0. For the function below, state the values of lim x → 0 − f ( x ) and It is not possible to approach 0 from the left, however, since x → 0 + f ( x ), if they exist. lim f ( x ) is not defined for any x < 0. Therefore, lim x → 0 − f ( x ) does not exist. J. Garvin — Limits and Their Properties J. Garvin — Limits and Their Properties Slide 13/23 Slide 14/23 l i m i t s l i m i t s Limit of a Function Limit of a Function We can define the limit of a function using left- and Example right-handed limits. Determine lim x → 2 f ( x ) for f ( x ) below, if it exists. Limit of a Function Given a function f ( x ), the limit as x → a exists if the left- and right-handed limits exist and are equal. Mathametically, x → a f ( x ) = L if lim lim x → a − f ( x ) = lim x → a + f ( x ) = L . Using this definition, the limit of f ( x ) as x → 0 in the previous example does not exist, since lim x → 0 − f ( x ) does not exist. A more formal definition of limits involving small quantities δ and ǫ is typically covered in first-year university courses, but Since lim x → 2 − f ( x ) = lim x → 2 + f ( x ) = 4, then lim x → 2 f ( x ) = 4. this definition will suit us for now. J. Garvin — Limits and Their Properties J. Garvin — Limits and Their Properties Slide 15/23 Slide 16/23 l i m i t s l i m i t s Limit of a Function Limit of a Function Example Example Determine lim x →− 1 f ( x ) for f ( x ) below, if it exists. Determine lim x → 1 f ( x ) for f ( x ) below, if it exists. Since x →− 1 − f ( x ) = 5 and lim x →− 1 + f ( x ) = 1, then lim lim x →− 1 f ( x ) Since lim x → 1 − f ( x ) = lim x → 1 + f ( x ) = 1, then lim x → 1 f ( x ) = 1. does not exist. J. Garvin — Limits and Their Properties J. Garvin — Limits and Their Properties Slide 17/23 Slide 18/23

l i m i t s l i m i t s Properties of Limits Properties of Limits Note that f ( x ) is discontinuous at x = 1, and that f (1) = 4. Limit Properties We will talk about this in more detail shortly. 1 lim x → a c = c if c is a constant. For now, just remember that the limit of a function as x 2 lim x → a x = a . approaches some value does not always need to have the 3 lim x → a [ cf ( x )] = c lim x → a f ( x ) for some constant c . same value as the function at that value. 4 lim x → a [ f ( x ) ± g ( x )] = lim x → a f ( x ) ± lim x → a g ( x ). While drawing a graph or creating a table of values is useful for simple functions, there are algebraic properties of limits 5 lim x → a [ f ( x ) g ( x )] = [ lim x → a f ( x )][ lim x → a g ( x )]. that make it possible to evaluate them without these tools. x → a f ( x ) lim f ( x ) Most of these are fairly easy to prove, if you are so inclined, 6 lim g ( x ) = x → a g ( x ) where lim x → a g ( x ) � = 0. lim but proofs are not provided here. x → a x → a [ f ( x )] n = [ lim x → a f ( x )] n where n is rational. 7 lim J. Garvin — Limits and Their Properties J. Garvin — Limits and Their Properties Slide 19/23 Slide 20/23 l i m i t s l i m i t s Properties of Limits Properties of Limits Example Example √ Use limit properties to evaluate lim x → 2 (3 x − 5). Use limit properties to evaluate lim 3 x + 1. x → 5 √ 1 x → 2 (3 x − 5) = lim lim x → 2 3 x − lim x → 2 5 (4) lim 3 x + 1 = lim x → 5 (3 x + 1) 2 x → 5 � 1 = 3 lim x → 2 x − lim x → 2 5 (3) � 2 = x → 5 (3 x + 1) lim (7) = 3(2) − 5 (1 and 2) � 1 � 2 = 3 lim x → 5 x + lim x → 5 1 (3 and 4) = 1 1 = [3(5) + 1] (1 and 2) 2 = 4 J. Garvin — Limits and Their Properties J. Garvin — Limits and Their Properties Slide 21/23 Slide 22/23 l i m i t s Questions? J. Garvin — Limits and Their Properties Slide 23/23