2.4 The Precise Definition of a Limit
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1 2.4 The Precise Definition of a Limit Reminders/Remarks: x 4 < 3 means that the distance between x and 4 is less than 3. In other words, x lies strictly between 1 and 7. So, x a < δ means that the distance between x and a is less than δ, that is, x lies within δ of a. Note: The inequality 0 < x a < δ means the distance between x and a is less than δ, but x a. Similarly, if we write f(x) L < 2, this means that f(x) lies within 2 of L. Consider the function f(x) = 2x + 4. We know x 3 (2x + 4) = 10. We said that the it means we can make f(x) as close to 10 as we want by getting x closer and closer to 3. How close to 3 does x need to be so that f(x) lies within 1 of the it value 10, i.e. f(x) 10 < 1? How close to 3 does x need to be so that f(x) lies within 0.1 of the it value 10, i.e. f(x) 10 < 0.1? How close to 3 does x need to be so that f(x) lies within an arbitrary number ǫ of 10, i.e. f(x) 10 < ǫ? This number that we re finding is denoted as δ. 1
2 ǫ,δ Definition of a Limit: Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say that the it of f(x) as x approaches a is L, and we write f(x) = L x a if for every number ǫ > 0 there exists a number δ > 0 such that whenever 0 < x a < δ, then f(x) L < ǫ. To prove a it using the definition of the it, there are two steps: 1. Do some scratchwork to determine a value for δ. 2. Show that this δ works. Example: Prove using the definition of the it that x 3 (2x+4) = 10. 2
3 Example: Prove using the definition of the it that (3x+7) = 1. x 2 Consider the function f(x) = x 2. We know x 2 x 2 = 4. Question: Given ǫ = 1, find a number δ so that if x 2 < δ, then x 2 4 < 1. Question: Given ǫ = 0.1, find a number δ so that if x 2 < δ, then x 2 4 <
4 Consider the function f(x) = 1 1 x2. We know =. We said that this means we can make f(x) as x 0 x2 large as we want by getting x closer and closer to 0. Definition: Left f be a function defined on some open interval that contains the number a, except possibly at a itself. Then x a f(x) = if for every number M > 0 there exists a number δ > 0 such that whenever 0 < x a < δ, then f(x) > M. Find a number δ so that if 0 < x < δ, then 1 x 2 > 100. Find a number δ so that if 0 < x < δ, then 1 x 2 > 10,000. Find a number δ so that if 0 < x < δ, then 1 > M, where M is some arbitrary number. x2 4
5 2.5 Continuity In Section 2.3 we saw that the it as x approaches a can sometimes be found by evaluating the function at a. If this is the case, then the function is continuous. Definition: A function is continuous at a number a if f(x) = f(a) x a Otherwise, we say the function is discontinuous at a, or that there is a discontinuity at a. In order for a function to be continuous at a number a: (1) f(a) must be defined. So a function will NOT be continuous anywhere it is undefined. (2) x a f(x) must exist. (The left-handed and right-handed its must both equal the same value.) (3) x a f(x) = f(a) Examples of discontinuities: Holes, vertical asymptotes, and jumps. A hole in a graph is also referred to as a removable discontinuity because if we wanted to, we could just redefine the function at that point to make it continuous. Removable discontinuities occur where the it exists at a (left and right its are equal), but is not equal to f(a). A vertical asymptote is referred to as an infinite discontinuity. A jump in the graph is referred to as a jump discontinuity. Jumps occur where the its from the left and right exist, but are not equal. 5
6 A function is continuous from the left at a number a if = f(a) and continuous from the x a f(x) right if x a +f(x) = f(a). A function is continuous if and only if it is continuous from both the right and the left. Examples: Determine where the functions below are discontinuous. State the type of discontinuity and explain why mathematically using its. Is the function continuous from the left or right there? (1) f(x) = x2 25 x 5 (2) f(x) = (x+3)(x 2) (x 2) 3 (3) f(x) = { x 2 4 if x 1 x+1 if x > 1 Fact: All polynomials are continuous everywhere! Fact: A rational function is continuous wherever it is defined, i.e. where the denominator is not 0. 6
7 2x 1 if x < 4 (4) f(x) = 6 if x = 4 x 2 9 if x > 4 (5) f(x) = 3x+1 if x < 2 x 2 5 x 1 if 2 x 3 x 3 25 x 2 if x > 3 What values of a, b, and c would make the following function continuous everywhere? ax 2 +bx+1 if x 1 3x+2a+b if 1 < x < 2 f(x) = c if x = 2 x 2 ax b if x > 2 7
8 If f and g are continuous at a and c is any constant, then the functions f +g, f g, cf, fg, and f g (where g(a) 0) are all continuous functions. The Intermediate Value Theorem: Suppose f is continuous on the closed interval [a, b] and let N be any number strictly between f(a) and f(b). Then there exists a number c in (a,b) such that f(c) = N. Example: Show that the equation x 3 +2x+2 = 0 has a root (solution) on the interval (1,2). Example: If f(x) = x 4 x 3 +3x 2 +2, show that there is a number c so that f(c) = 3. 8
9 2.6 Limits at Infinity; Horizontal Asymptotes Up to this point, we have dealt with its as x approaches some number a. Now, we examine its as x approaches or. These are called its at infinity. Let f be a function defined on some interval (a, ). Then, f(x) = L x means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large. Definition: If f(x) = L or f(x) = L, then f(x) has a horizontal asymptote at y = L. x x 1 x x = 2 x x = 3 = x ± x 2 x (x3 x 2 ) x (x3 x 2 ) Evaluate the following its: 2x 3 +x 2 1 x 5x 3 7x+2 2x 3 +x 2 1 x 5x 3 7x+2 9
10 4x 2 2x+3 x 5 3x 4x 2 2x+3 x 5 3x x x 2 (x 8) (x 2 +1)(x 2 3) x x 2 (x 8) (x 2 +1)(x 2 3) x 9x x+2 x 9x x+2 10
11 x 2 x x x x 2 +5x+x x Find all horizontal asymptotes of the function f(x) = 5 4x3 25x
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