TOPIC : 8.0 LIMITS AND CONTINUITY 8.1 LIMITS 8.2 ASYMPTOTES 8.3 CONTINUITY
LECTURE 1 OF 4 TOPIC : 8.0 LIMITS AND CONTINUITY 8.1 LIMITS
State limit of a function f(x) as x approaches a given value a , State the basic properties of limit. lim ( ) . x a f x L Find when and by the following methods: i. factorization ii. multiplication of conjugates lim ( ) lim ( ) x a x a f x g x lim ( ) 0,x a g x Find one-sided limits in: ii f xLx a i f x M . lim( ) x a .lim ( ) Determine the existence of the limit of a function lim f (x) lim f (x) x a x a lim ( ) 0 x a f x OBJECTIVES At the end of this topic, students should be able to:
8.1 CONCEPTS OF LIMITS In Mathematics, limit of a function refers to a number, L which f(x) approaches, as x approaches a certain value, a. In other words, we look at the ‘behaviour’ of f(x) as x comes closer and closer to a; x=a f(x)=L 0 f(x) x as x a , f (x) LAs x approaches a from the left and right, f(x) will approach a certain number , L .
DEFINITION : THE LIMIT OF A FUNCTION AT ANY GIVEN POINT If f(x) becomes closer and closer to a single number L , as x approaches a from either side, then lim f x L x a ( ) which is read as : “the limit of f(x) as x approaches a is L”
NOTATION x a x approaches a from the right : x a x approaches a from the left : x a x approaches a from both sides : x a x a x a x a x a
x=3 f(3)= 6 f(x)=x+3 x f(x) 0 As x 3 - , f(x) 6 As x 3 + , f(x) 6 ( 3) 3 lim x x 6 As x approaches 3 from the left,f(x) will approach, 6 As x approaches 3 from the right,f(x) will approach, 6
NOTES x a x a If f ( x ) f ( x ) L,lim lim x a x a If f ( x ) f ( x ) , lim lim x a lim f ( x ) L x a lim f ( x ) does not exist
2 0 lim( 1) x x x - 1 - 0.5 - 0.25 - 0.001 0 0.001 0.25 0.5 1 f(x) 2 1.25 1.0625 1.000001 ? 1.000001 1.0625 1.25 2 As x approaches 0 from left f(x) is getting closer to 1 As x approaches 0 from right f(x) is getting closer to 1 2 0 ( 1) 1 x li m x 2 0 ( 1) x li m x 2 0 ( 1) x li m x 1 1 x 0- , f(x) 1 x 0+ , f(x) 1 (From table)
x f(x) f(x)=x2+1 0 1 x<0 x>0 ( 1) 2 0 lim x x ( 1) 2 0 lim x x 1 1 ( 1) 1 2 0 lim x x x=0 , f(0)=1 (From graph)
PROPERTIES OF LIMITS 1 x a . limc c 2 x a .lim x a 3. n n x a li m x a 4 x a x a . c f ( x ) c f ( x ) lim lim 5 n n x a x a . li m f ( x ) li m f ( x ) 6. ( ) ( ) ( ) ( ) x a x a x a li m f x g x li m f x li m g x 7 x a x a x a . li m f ( x ) g( x ) li m f ( x ) li m g( x ) 8. ( ) ( ) , ( ) 0n n x a x a x a li m f x li m f x where n Z li m f x
Direct substitution method The limits of certain functions may be evaluated by substituting x=a in f(x) Such that ; lim f (x) f (a) x a Computation of limits
EXAMPLE 1: Evaluate the following : ( ) lim24 b xx ( ) limln4 t3 a 1 1 ( ) lim2 x x c e ( ) lim[ln( 1)5]2 d xx 3 2 1 11( ) lim xxf x ( ) lim 2 3 e x x
3 ( ) 4 t a li m l n l n4 4 ( ) 2 x b li m x 42 2 1 1 ( ) 2 x x c li m e 1 1 2 e 2 2(1) 2 0 e SOLUTION:
x 3 (e) lim x 2 3 2 1 2 3 x 1 x 1 (f ) lim x 1 2 3 x 1 x 1 lim x 1 3 2 1 2 2 1 5 x ( d ) li m ( l n( x ) ) l n( ) 2 1 5 l n1 5 5
Direct substitution method The limit of a rational function can be found by direct substitution when the denominator is different from zero. Computation of limits: Limit of the Rational Function ( ) lim ( ) f x x a g x ( ) ( ) f a g a provided ( ) 0g a
Find the 3 2 2 4 lim x 3 x x 3 2 2 4 lim x 3 x x 2 8 4 lim x 4 3 4 SOLUTION: EXAMPLE 2:
Computation of limits: Limit of the Rational Function ( ) lim ( ) f x x a g x 0 0 If both f(x) and g(x) are functions with 0 and , x a lim f ( x ) 0x a lim g( x )this is called : the indeterminate form 1. Factorization (for polynomials) 2. Multiplication of Conjugates (for surds) 3. Define Modulus ( for |x| ) then this can be solved by either : Direct substitution method???
Find the 4 8 lim 2 3 2 x x x 3 2 2 8 lim x 4 x x 2 2 2 4 lim x 2 x x x 2 2 ( 2)( 2 4) lim ( 2)( 2) x x x x x x 2 2 2(2) 4 2 2 3 4 12 SOLUTION: 1. Factorization Method EXAMPLE 3: 0 0 the indeterminate form 3 2 2 (2) 8 lim x (2) 4
2 2lim 2 2x x x 2 2 2 2 . x x 2 ( 2)( 2 2) lim ( 2) x x x x 2 lim 2 2 x x 4 2 2 lim 2 2 x x x 22 2 Find the limit SOLUTION: EXAMPLE 4: 2. Multiplication Of Conjugate Method 0 0 the indeterminate form 2 2 2 lim 2 2 2 x
EXAMPLE 5: Does the exist ? x x and x x x x 0 0 lim lim x x x 0 lim Find
x x x 0 lim x x x 0 lim 1 x x x 0 lim x x x 0 lim 1 0 0 lim lim x x x x x x x x x 0 lim Therefore does not exist. , 0 , 0 x x x x x SOLUTION:
Case Study
3 6 Therefore the limit of this function is called a two – sided limit -3 3 9 ( ) 2 x x f x 3 lim ( ) 6 x f x 3 2 3 9 lim x 3 x x 3 ( 3)( 3) lim x ( 3) x x x 3 lim( 3) x x 6 Df is ( ,3) (3, ) Case Study: two – sided limit 2 3 9limx 3xx 3 ( 3)( 3)limx ( 3)x xx 3 lim( 3)x x 6 2 3 9 lim x 3 x x 2 3 9lim 6x 3x x Since
ONE-SIDED LIMIT A one-sided limit can either be a left–hand limit or a right-hand limit The limit of f as x approaches a from the left is L f x L x a lim ( ) x y y = f(x) a L Left-hand limit The limit of f as x approaches a from the right is M f x M x a lim ( ) y=f(x) x y a M Right-hand limit
The Existence of a Limit x a x a If f ( x ) f ( x ) L, lim lim x a x a If f ( x ) f ( x ), lim lim x a lim f ( x ) L x a lim f ( x ) does not exist (Limit exist at x=a) The Relationship between One-Sided Limits and Two-Sided Limits
(a) if (b) if SOLUTION: EXAMPLE 6: Determine the limits of the following functions: 0, 4 , 4 ( ) 2 x x x f x 1 , 0 , 0 ( ) 2 x x x x f x lim ( ) 4 f x x lim ( ) 0 f x x
SOLUTION: a) 16 4 y x 2 4 4 lim ( ) lim 16 x x f x x 2 4 4 lim ( ) lim 16 x x f x x 0, 4 , 4 ( ) 2 x x x f x 4 lim ( ) 16 x f x lim ( ) 4 f x x
SOLUTION: (b) 1 y x f x doesnot exist x lim ( ) 0 1 2 1 , 0 , 0 ( ) 2 x x x x f x 2 0 0 lim ( ) lim 0 x x f x x 0 0 lim ( ) lim1 1 x x f x x lim ( ) 0 f x x
0 1 1 lim x x x (1 1 ) (1 1) x x 0 1 1 lim (1 1 ) x x x x 0 1 lim (1 1) x x 2 1 1 1 1 Find the limit SOLUTION: 0 1 1 lim x x x EXERCISE 1:
h 0 , lim f(x+h)-f(x)If f(x) = x find hh 0 f ( x h ) f ( x ) lim h h 0 x h x lim h SOLUTION: If f ( x ) x f ( x h ) x h EXERCISE 2:
h 0 x h x x h x h x h x lim . 0 lim h x h x h x h x x h x h 1 lim 0 2 x 1 DON’T EXPAND!!!
2 , 4 ( ) 2 3 , 4 x x Given the function f x x x 8 0 11 x y 4 lim x x 2 4 (2 3) 4 lim x x 11 4 4 4 x x li m f ( x ) li m f ( x ) li m f ( x ) does not exist 8 f(x)=2x f(x)=2x+3 x<4 x=4 x>4 x<4 x>4 EXERCISE 3: SOLUTION: 4 ( ). x Find li m f x
this is called : the indeterminate form If direct substitution is failed, thus we should go for factorizations method in order to find the limit. x 1 x lim 1 x 1 2 ( 1)( 1) ( 1) x x x lim x 1 lim x 1 ( 1) x = 2 0 0 x 1 x lim 1 x 1 2 EXERCISE 4: SOLUTION: x1xlim1x 1 2 Find
If DIRECT SUBSTITUTION method FAILED.. 0 0 getting the indeterminate form x 1 x lim 1 x 1 2 2 x x lim x } 1. Factorization 2. MULTIPLy by its CONJUGATE ( ) lim ( ) f x x a g x both f(x) and g(x) are polynomial funtions either one of f(x) and g(x) are funtions in surds form For CONCLUSIONS