LIMIT AND CONTINUITY (NPD)

3.1 Limit Function at One Point An intuitive Let This function is not defined at x = 1 ( dividing by 0). We still ask, what happening to f(x) as x approaches 1. The following table shows some values of f(x) for x near 1 x 0.9 0.99 0.999 0.9999 1 1.0001 1.001 1.01 1.1 f(x) 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1

Geometrically 1 º The result in the table and graphic suggest that the values of f(x) approaches 2 as x approaches 1. f(x) 2 We denote this by writing f(x) x x we read : the limit as x approaches 1 of is 2 Definition (intuitive) : Means that when x is near but different from c, then f(x) is near L.

Example 1: a. b. c. d. If values of x approach 0 we get value of f(x): x 1 -1 1 -1 ? If x approaches 0, sin(1/x) does not approach to any number. Thus limit does not exist.

Right- and Left-Hand Limits (one-sided limits) If x approaches c from the left side (from number on which is smaller than c), we get the left-hand limit, denoted by x c

Right- and Left-Hand Limits (one-sided limits) If x approaches c from the right side (from number on which is greater than c), we get the right-hand limit, denoted by c x

Example 4

Example 5 Let a. Evaluate b. Evaluate c. Evaluate d. Sketch graph of f(x)

Answer Because the formula of f(x) change at x = 0, then we find the one-sided limit at point x = 0. b. Because the formula of f(x) change at x = 1, then we find the one-sided limit at point x = 1. c. The formula of f(x) does not change at x = 2, thus

d. 3 at x = 1 limit does not exist º 1 For x 0 For 0 < x < 1 For x 1 f(x) = x Graph: parabola Graph: straight line Graph: parabola

2. Find the value of c so that has a limit at x = -1 Solution : f(x) has a limit at x = -1 if the limit from the right = the limit from the left limit exist if 3 + c = 1 - c C = -1

EXERCISES A. For the function f graphed below Find: 1. 4. f(-3) f(-1) 2. 5. 6. f(1) 3.

B. 1. Let Evaluate b. Does exist? if it limit exist evaluate 2. Let , evaluate a. b. c. 3. Let , evaluate a. b. c.

Theorem Let then

The Squeeze Theorem (Sandwich) Let for all x in some open interval containing the point c and then Example 6: Evaluate Because and then

Example 7 : x  0 equivalent with 4x  0

Exercises Evaluate: 1. 2. 3. 4.

The Precise Definition of a Limit

In general

Infinite Limits and Limits at Infinity a. Infinite Limits Remark : g(x)  0 from upward mean g(x) approaches 0 from positive value of g(x). g(x)  0 from downward mean g(x) approaches 0 from negative value of g(x).

Example 8: Evaluate a. b. c. Solution a. , g(x) = x - 1 approaches 0 from the downward thus b. , approaches 0 from the upward thus

If x approaches from the right then sin(x) approaches 0 from downward Because f(x)=sinx and x If x approaches from the right then sin(x) approaches 0 from downward thus

Limits at Infinity

Example 9 : Evaluate Solution = 1/2

Example 10: Evaluate Solution = 0

Example 11 : Evaluate Solution: If x  , It is form ( )

Exercise Evaluate 1. . . 2. 3. 4. 5. 6.

Continuity A Function f(x) is said continuous at x = c if (i) f(c) is defined (ii) (iii) If one or more of these 3 conditions fails, then f is discontinuous at c and c is a point of discontinuity of f (i) f(c) does not exist º c f is not continuous at c

(ii) Left-hand limit ≠ right-hand limit , Thus limit does not exist at x = c c f is not continuous at x = c f(c) ● (iii) f(c) is defined L º exist but c f is not continuous at x = c

f(c) is defined (iv) exist f(c) c f(x) is continuous x=c Removable discontinuity For case (i) discontinuity can be removed by define f(c) = limit of function at c

Example Determine whether f ,g ,and h are continuous at x = 2 a. b. c. Solution: a. f is not defined at x = 2 (0/0) f is not continuous at x = 2 b. g(2) = 3, g is not continuous at x=2

c. h is continuous at x = 2

Continuity From the Left and the Right A function f(x) is called continuous from the left at x =a if A function f(x) is called continuous from the right at x = a if f(x) is continuous at x = a if f(x) is continuous from the left and right Example : Find the value of a so that f(x) will be continuous at x = 2

Solution : f(x) is continuous at x = 2, if : 1. f is continuous from the left at x = 2 2 + a = 4a – 1 -3a = -3 a = 1 2. f is continuous from the right at x = 2 trivial

Problems 1. Let Determine whether f is continuous at x = -1 2. Find the value of a + 2b so that f(x) will be continuous everywhere 3. Find the values of a and b so that f(x) will be continuous at x = 2

Continuity on a Closed Interval A function f(x) is said to be continuous on closed interval [a,b] if 1. f(x) is continuous on ( a,b ) 2. f(x) is continuous from the right at x = a 3. f(x) is continuous from the left at x = b

Theorem A Polynomials are continuous function A rational function is continuous everywhere except at the points where the denominator is zero Theorem B Let , then f(x) is continuous everywhere if n is odd f(x) is continuous for positive number if n is even Example: In which x s.t. is continuous ? From theorem, f(x) is continuous for x – 4 > 0 or x > 4. Thus f(x) is continuous at [4, ) f(x) is continuous from the right at x = 4

Example Show that there is a root of the equation 𝑥 3 −𝑥 −1=0 between 1 and 2

Problems A. Find the points of discontinuity, if any 1. 3. 2. B. Find the points where f(x) are continuous 1. 2.