TBF General Mathematics-I Lecture – 3 : Limits and Continuity Prof. Dr. Halil İbrahim Karakaş Başkent University

Limits. Let f be a function and let c, L  ℝ. Assume that there is an open interval (a (a, b)b) containing c such that f is defined in (a (a, b) \ {c}. f may or may not be defined at c. In any case, the following question is of considerable impotance. L: How does f(x) change as the variable x gets closer and closer to c?c? “x “x approaches c”c” x  c x |1-x||1-x| x  0 |1-x|  0 L 1 : As x approaches c, f(x) approaches L. L 2 : f(x) becomes sufficiently close to L if x is sufficiently close to c.c. L 3 : If the distance from x to c aproaches zero, then the distance from f(x) to L approaches zero. The question L may have answers similar to the following:

x   If x  c and x < c, then we say that x approaches c from the left and we write xc– xc–. If x  c and x > c, then we say that x approaches c from the right and we write xc+ xc+. x  c– x  c– c c x  c+ x  c+   x

L: How does f(x) change as the variable x approaches c?c? L 1 : As x approaches c, f(x) approaches L. L 2 : f(x) will be sufficiently close to L if x is sufficiently close to c.c. L 3 : If the distance from x to c aproaches zero, then the distance from f(x) to L approaches zero. If any of the sentences L 1, L2 L2 or L 3 is valid, then L is called the limit of f as x approaches c. In this case one writes or as x y (0,0) (c,L) c L x f(x)f(x) x (x,f(x)) f(x)f(x)

x y (0,0) (c,L) c L x f(x)f(x) x (x,f(x)) f(x)f(x) or Example. x y (0,0) 1/

Definition. Let f be a function and let c, L  ℝ. Assume that ther is an open interval (a (a, b)b) containing c such that f is defined in (a (a, b) \ c. If for every  > 0 there exists  >0 such that then L is called the limit of f as x sapproaches c.c. or In this case one writes The condition in the definition can also be expressed as: For each  > 0 there exists  >0 such that whenever one has (c,L) c x y (0,0) c+  L c-  x (x,f(x)) L-  f(x)f(x) L+  “  -  definition”

Given  > 0. Take  =  /

y x (0,0) 1 2 For x  1, We say that f has no limit as x  c if there is no L such that y x (0,0) No limit! 2

(0,0) x y x y does not exist!

Left Limit limit of f as x approaches c from the left x y (0,0) (c,L) c L x (x,f(x)) or

(5,0) y x (0,0) (0,0) x y 0

x y x y no limit!, then we say that f does not have a limit as x aproaches c from the left. If there is no L such that

Definition. Let f be a function and let c, L  ℝ. Assume that ther is an open interval (a (a, b)b) containing c such that f is defined in (a (a, b) \ {c}. If for every  > 0 there exists  >0 such that then L is called the limit of f as x sapproaches c from the left. or We have “ “ -  definition ” for left limits too: (c,L) c x y (0,0) L c-  x L-  L+  f(x)f(x) (x,f(x))

Right Limit. limit of f as x approaches c from the right or x y (0,0) (c,L) c L x f(x)f(x) (x,f(x)) y x (0,0)

y x x y 0

, then we say that f does not have a limit as x aproaches c from the right. If there is no L such that (0,0) x y x y does not exist!

We have “ “ -  definition ” for right limits too: Definition. Let f be a function and let c, L  ℝ. Assume that ther is an open interval (a (a, b)b) containing c such that f is defined in (a (a, b) \ {c}. If for every  > 0 there exists  >0 such that then L is called the limit of f as x sapproaches c from the right. or (c,L) c x y (0,0) L c+  x L-  L+  f(x)f(x) (x,f(x))

Limit Left Limit Right Limit does not exist

For a function f, defined by an equation y = f(x), we have Sketch a graph for f.f. x y (-1,2) (1,3) (1,-2) (1,2) (2,0) (0,0) y = f(x) (0,1)

Properties of Limits. Let f and g be two functions; c, k, L, M  ℝ;ℝ; Then All these properties hold true if x  c is replaced with x  c+ c+ or x  c -.

Continuity. Look at the graph of each of the following functions near x = 2. x y (0,0) y x 2 4 y x Definition. f is said to be continuous at x = c if the following three conditions hold continuous at x = 2 not continuous at x = 2 not continuous at x = 2

Definition. f is said to be continuous at x = c if the following three conditions hold If f is not continuous at x = c, then f is said to be discontinuous at x = c. Here is an example of another discontinuity: y = f(x) c x y (0,0) f(c)f(c) L (c,f(c)) (c,L)(c,L) A concrete example: x y (0,0)

Definition. Let a, b  ℝ, a < b. If a function f is continuous at c for all c  ( a,b ), then f is said to be continuous on the interval (a, b) b). f is continuous on the intervals (- ,-1), (-1,0), (0,1), (1, )) x y (-1,2) (1,3) (1,-2) (1,2) (2,0) (0,0) y = f(x)

Theorem(Intermediate Value Theorem). Let f be a function which is continuous on the interval (a, b) and assume that f (x)  0 for all x  (a, b). Then either f(x) > 0 for all x  (a, b) or f (x) < 0 forall x  (a, b).b). y x (0,0) a b Definition. A number x is called a partition number of a function f if f is discontinuous at x or f(x) = 0.0. y = f(x)

Example. The function p defined by p(x)=x–2 has one single partition number: x = 2. x-2 0 x --  sign chart Example. x The function defined by p(x)=x 2 –1 has two partition numbers: x = and x = 1. x --  Example. Partition numbers of : x =, 1 and x-2 0 x - 1 x --  x

x y x-2 0 x - 1 x --  x

Continuity on one side. If, then f is saidto be continuous on the left at x = c. y x (0,0) y x y x continuous on the right at x = continuous on the left at x = 1 continuous on the right at x = 0 continuous neither on the right nor on the left at x = 0 1 If, then f is saidto be continuous on the right at x = c.

Infinite Limits and Vertical Asymptotes. Let f be a function and c ℝ ℝ where f may or may not be defined at c but it is defined at all other numbers in an interval (a, b) containing c. If f (x) gets larger and larger as x approaches (from both sides) to c, then we say that f has infinite limit as x  c or f(x) diverges to infinity as x  c. We epress this fact by writing or Similarly, if f (x) (x) is negative and | f (x) | gets larger and larger as x approaches (from both sides) to c, then we say that f has minus infinite limit as x  c or f(x) diverges to minus infinity as x  c. We epress this fact by writing or y x (0,0) c y x c

“ “ -  definition” for infinite limits: Definition. Given a function f and a number c  R; assume that f is defined at all numbers in an open interval containing c except perhaps at c. If for every  > 0 one can find a  >0 such that then we say that f has infinite limit as x  c or f(x) diverges to infinity as x  c. This fact is expressed as or then we say that f has minus infinite limit as x  c or f(x) diverges to minus infinity as x  c. This fact is expressed as Similarly, if for every  > 0 one can find a  >0 such that or

y x (0,0) 2 y x 2

y x c It is clear now how to define “one sided infinite limits” y x (0,0) c y x c y x c

y x y x 1

If any one of holds, then the line x = c is said to be vertical asymptote to the graph of f. In this case, we also say that the function f has a vertical asymptote, x = c. From our previous work, the line x = 1 is vertical asymptote to the graph of x = 1 is also vertical asymptote to the graph of The line x = 2 is vertical asymptote to the graph of both and has another vertical asymptote: x = -2, becouse

Limits at Infinity. Let c be real number and f a function defined on the interval (c,  ). If f(x) f(x) approaches a number b as x increases indefinitely, then b is said to be the limit of f as x diverges to infinity. This fact is expressed as or y x (0,0) b y x b Similarly, let c be real number and f a function defined on the interval (- , c). If f(x) f(x) approaches a number b as x decreases indefinitely, then b is said to be the limit of f as x diverges to minus infinity. This fact is expressed as or y x (0,0) b y x b

“ “ -  definition” for limits at infinity: Definition. Let f fbe a function; b, c  ℝ. Assume that f is defined on the interval (c, ) ). If for every  > 0 one can find a  > 0 such that then, b is said to be the limit of f as x diverges to infinity. or Similarly, assume that f is defined on the interval (- , c). If for every  > 0 one can find a  > 0 such that then, b is said to be the limit of f as x diverges to infinity. or

Examples. Let us note that all the limit properties we stated before hold for limits at infinity.

Horizontal Asymptotes. If then the line y = b is said to be an horizontal asymp- tote to the graph of f. Horizontal asymptotes of or and  y = 2 horizontal asymptote. Vertical asymptotes of and  x = 1 vertical asymptote.

Infinite Limits at Infinity. Let c be a real number and f a function defined on the interval (c,  ). If f (x) increases indefinitely as x increases indefinitely, then we say that the limit of f as x   is infinity and we write or Similarly, if If f (x) decreases indefinitely as x increases indefinitely, then we say that the limit of f as x   is minus infinity and we write or We believe that the reader can easily understand the meaning of and

Examples. y x (0,0) y x y x y x

Examples.