2.2 The Limit of a Function


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1 2.2 The Limit of a Function Introductory Example: Consider the function f(x) = x is near 0. x f(x) x f(x) x x+4 2. Examine the values of the function when We can see that as the values of x get closer to 0 from both sides, the values of f(x) are getting closer to 4. x This is written mathematically as: = 4 x 0 x+4 2 We write f(x) = L x a read as the it of f(x) as x approaches a is L if we can make the values of f(x) arbitrarily close to L (as close to L as we want) by letting x get closer and closer to a, but not equal to a. 0 Note that f(0) is undefined in the above example since f(0) = = 0 0. When dealing with its, we are examining values as x approaches a, but not equal to a. We ll see later that the value of the function at a may or may not equal the value of the it. For all three figures below: LeftHanded Limit: x a f(x) RightHanded Limit: x a +f(x) The it exists if and only if the lefthanded and righthanded its both exist and are equal. x a f(x) = L if and only if = L and x a f(x) x a +f(x) = L 1
2 1 Infinite Limits: Calculate x 0 x 2 x f(x) x f(x) , , , 000, , 000, , 000, , 000, 000 We can see that as the values of x approach 0 from both sides, that f(x) gets larger and larger without 1 bound. In this case, we say that x 0 x 2 =. 1 Example: Calculate x 0 + x, x x 0 1 1, and x 0 x Definition: The line x = a is a vertical asymptote if the it from the left, right, or both is or. Example: Consider the graph of f below. Find the indicated its. x 4 f(x) x 4 +f(x) f(x) x 4 x 2 f(x) x 2 +f(x) f(x) x f(x) x 1 4 f(x) x 6 2 x 3 +f(x) x 3 f(x) f(x) x f(x) x What are the vertical asymptotes of f(x)? 8 2
3 Asymptotes vs Holes When the it of a function at x = a is of the form nonzero 0, then there is a vertical asymptote at x = a. Why? When the it of a function at x = a is of the form 0 0 we say the it is indeterminate. Two things could be happening. There is either a vertical asymptote or a hole in the graph at x = a. If after using algebra the it simplifies to nonzero 0, then there is a vertical asymptote. If the it simplifies to a number, then there is a hole. Find all vertical asymptotes of the function f(x) = at all such values of x. x 2 x 2 6x+8 and calculate the left and righthand its Calculate the following its: x 1 x 5 x+5 x 1 x 5 + x+5 x 1 x 5 x+5 x 3 x 3 (x 6)(x 3) 2 3
4 2.3 Calculating Limits Using the Limit Laws Limit Laws: Also found on pp of your textbook. Suppose f(x) and g(x) exist and that c is x a x a any constant. Then: 1. x a (f(x)±g(x)) = x a f(x)± x a g(x) 2. x a cf(x) = c x a f(x) 3. x a f(x)g(x) = x a f(x) x a g(x) f(x) 4. x a g(x) = f(x) x a g(x) x a 5. x a (f(x)) n = provided that x a g(x) 0 ( ) n f(x) x a 6. x a n f(x) = n x a f(x). If n is even, then we must have that x a f(x) > 0 In particular, 7. x a c = c 8. x a x = a 9. x a x n = a n Given that x 3 f(x) = 16, calculate x 3 (x 2 4)f(x) f(x)+x+2 If f is a polynomial or rational function and a is in the domain of f, then x a f(x) = f(a). x 2 (3x2 +5x+1) 4 4
5 We saw in the previous section what happens when only the it of the denominator is 0. There is an infinite it which yields a vertical asymptote. If the its of both the numerator AND denominator are 0, the it is indeterminate, so you must USE ALGEBRA to determine the it. Some methods are expanding, factoring, or multiplying by the conjugate of a radical. x 2 x 12 x 4 x 2 16 (h 4) 2 16 h 0 h t 2 t 2 (t 2) 3 x 1 x+3 2 x 1 5
6 For vector functions, if r(t) =< f(t),g(t) >, then r(t) = f(t),g(t) t a t a t a provided the its of the component functions exist. 2(t 2) 1 2 t Calculate r(t), where r(t) =, t 3 t 3 t 3 9 t(t 3) Find a number a so that the following it exists and then find the value of the it. x 2 +ax+a+5 x 3 (x+3)(x 5) 6
7 Let f(x) = x if x < 0 x 2 if 0 x < 2 8 x if 2 x < 5 2 if x = 5 x 2 if x > 5 Calculate x 0 f(x), x 2 f(x), and x 5 f(x) or explain why the it does not exist. { Recall the definition of the absolute value function: x = Calculate x 0 x 2 +x x or explain why the it does not exist. 7
8 Calculate x 3 x 3 6 2x or explain why the it does not exist. Squeeze Theorem: If g(x) h(x) f(x) for all x in an interval that contains a (except possibly at a) and g(x) = f(x) = L x a x a then x a h(x) = L Example: If 4x 2 f(x) x 2 +2 for 0 x 3, find x 2 f(x). ) Example: Find x sin( 2 1 x 0 x. 8
9 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 lefthanded and righthanded 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. 9
10 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. 10
11 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 and b would make the following function continuous everywhere? What are the values of f(x) and f(x)? x 3 x 1 ax 2 +bx+1 if x 3 x f(x) = 2 +6x 7 if 3 < x < 1 x 1 4x+a+2b if x 1 11
12 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. 12
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