2.2 The Limit of a Function (Part 1) Intuitive Denition of 3 Limit We begin our quest to understanding limits, as our mathematical ancestors did, by using intuition. 1. Consider the functions shown below: X 24 x2 1'06) = x24 2 behaves around x = 2. x- a. Look at how the function f (x) = As x approaches 2 from the right side, the value of y = f (x) approaches As x approaches 2 from the left side, the value of y = f (x) approaches | 2| behaves around x = 2. b. Look at how the function g(x) = : As x approaches 2 from the right side, the value of y = g(x) approaches As x approaches 2 from the left side, the value of y = g(x) approaches Intuitive denition of the limit: We can think of the limit of a function at a number a as being the real number L that the functional values approach as the x values approach a. Limit Denition: Suppose the function x] is defined for all x near a except possibly at a. Iff(x) is arbitrarily close to L (as close to L as we like] for all x sufficiently close (but not equal] to a, we write lim f(x) = L x>a and say "the limit off(x) as x approaches a equals L." sin(x) 2. Evaluate the lim 0 x using the table of functional values below: xQ x sin(x) 26 0.1 0.998334166468 0.01 0.999983333417 0.001 0.999999833333 0.001 0.999999833333 0 Undefined 0.0001 0.999999833333 0.001 0.999999833333 0.01 0.999983333417 0.1 0.998334166468 lim 51n(x) = _ an x VX+1Z 3. Evaluate the lim using a table of functional values. x73 x3 x Wz L 3 Undefined 4. Using the graph of g(x), we can find lim g(x). X - - 1 ' YA g(x) lim g(x) = 5. Using the graph of g(x), we can find lim h(x). YA h(x) lim h(x) =6. Find the following limits using the graph provided. f (x ) = 2 a. lim 2 = X -3 b. lim 2 = x- -2 lim 2 = X- 0 -2 d. lim x = x-4 e. lim x = x--6 f. lim x = x-+1 Theorem: Two Important Limits Let a be a real number and c be a constant i . lim c = c x - a 11 lim x = a x-a7. Find the following: a. lim 5 = x-2 b. lim x = x-999 C. lim 7 = x-100 d. lim 2x + 10 = x-11 8. Consider the graph of the function y = f (x) shown here. a. lim f (x) = x- -8 ' 6+ 5 b . f ( - 8 ) = 4 3. c. lim f (x) = X -8 - 10 -8 -6 4 -2 0 2 4 6 8 10 -1- d. lim f (x) = -2 x-8 -3 -4 -5- e. f (8) = -6- -7