d Limits at infinity and infinite limits i E 2 Lectures a l l - PowerPoint PPT Presentation

Section 2.6 d Limits at infinity and infinite limits i E 2 Lectures a l l u d b Dr. Abdulla Eid A . College of Science r D MATHS 101: Calculus I Dr. Abdulla Eid (University of Bahrain) Infinite Limits 1 / 29

d 1 Finite limits as x → ± ∞ . i E 2 Horizontal Asympotes. a l l 3 Infinite limits. u d 4 Vertical Asymptotes. b A . r D Dr. Abdulla Eid (University of Bahrain) Infinite Limits 2 / 29

Motivation Example Consider the function f ( x ) = 1 x . The graph of the function is y d i E a x l l u d b A . r D Question: What happen if x is sufficiently large number (i.e., x approaches ∞ ) ? In other words, what is lim x → ∞ 1 x ? From the graph we can easily see that 1 1 lim x = 0 and lim x = 0 x → ∞ x →− ∞ Dr. Abdulla Eid (University of Bahrain) Infinite Limits 3 / 29

Continue... Arithmetic at infinity: d i E ∞ + ∞ = ∞ . 1 a 2 k · ∞ = ∞ ( k > 0). l l u 3 k · ∞ = − ∞ ( k < 0). d b 1 ± ∞ = 0. A 4 ∞ ∞ = ? . (Calculus 1). 5 . r ∞ − ∞ = ? , 0 · ∞ = ? , 1 ∞ = ? . (Calculus 2) D 6 Dr. Abdulla Eid (University of Bahrain) Infinite Limits 4 / 29

Finding the limit of a rational function d f ( x ) i To find the limit lim x →± ∞ g ( x ) , we have E 1 Substitute directly by x = ± ∞ in f ( x ) a g ( x ) . If you get a real number or l l u ± ∞ , then that is the limit. d 2 If you get undefined values such as 0 b 0 or ∞ ∞ , we take the highest power A of x in the numerator and the highest power of x in the denominator . as common factor and we proceed. r D Dr. Abdulla Eid (University of Bahrain) Infinite Limits 5 / 29

Example 1 Find 3 x 2 − x − 2 lim 5 x 2 + 4 x + 1 x → ∞ Solution: Direct substitution gives d i 3 ( ∞ ) 2 − ( ∞ ) − 2 E undefined! 5 ( ∞ ) 2 + 4 ( ∞ ) + 1 a l l u 3 x 2 − x − 2 d 3 x 2 − x − 2 b 5 x 2 + 4 x + 1 = lim lim 5 x 2 + 4 x + 1 A x → ∞ x → ∞ x 2 � 3 − 1 x − 2 � . x 2 r = lim D 5 + 4 x + 1 x 2 � � x → ∞ x 2 3 − 1 x − 2 � � x 2 = lim 5 + 4 x + 1 � � x → ∞ x 2 = ( 3 − 0 − 0 ) ( 5 + 0 + 0 ) = 3 5 Dr. Abdulla Eid (University of Bahrain) Infinite Limits 6 / 29

Exercise 2 Find 5 x 2 − 2 x + 1 lim 9 x 2 x → ∞ d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Infinite Limits 7 / 29

Example 3 Find 3 x + 7 lim x 2 − 2 x → ∞ Solution: Direct substitution gives d i 3 ∞ + 7 E ( ∞ ) 2 − 2 = ∞ undefined! a ∞ l l u 3 x + 7 3 x + 7 d x 2 − 2 = lim lim b x 2 − 2 x → ∞ x → ∞ A 3 + 7 � � x x = lim . r 1 − 2 x 2 � � x → ∞ D x 2 3 + 7 � � x = lim 1 − 2 � � x → ∞ x x 2 ( 3 + 0 ) = ∞ ( 1 − 0 ) = 0 Dr. Abdulla Eid (University of Bahrain) Infinite Limits 8 / 29

Example 4 Find x 3 − 8 lim 2 x 2 + 1 x → ∞ Solution: Direct substitution gives d i E ( ∞ ) 3 − 8 2 ( ∞ ) 2 + 1 = ∞ undefined! a l ∞ l u d b x 3 − 8 x 3 − 8 A lim 2 x 2 + 1 = lim 2 x 2 + 1 x → ∞ x → ∞ . r x 3 � 1 − 8 � D x 3 = lim 2 + 1 x 2 � � x → ∞ x 2 1 − 8 � � x x 3 = lim 2 + 1 � � x → ∞ x 2 = ∞ Dr. Abdulla Eid (University of Bahrain) Infinite Limits 9 / 29

Exercise 5 Find 5 x 3 − 2 x + 1 lim 9 x 8 + 8 x → ∞ d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Infinite Limits 10 / 29

Example 6 Find √ 3 x 2 + 1 lim 3 x − 5 x → ∞ Solution: Direct substitution gives d i E 3 ( ∞ ) 2 + 1 � = ∞ lim undefined! a 3 ( ∞ ) − 5 x → ∞ l ∞ l u √ √ d 3 x 2 + 1 3 x 2 + 1 b lim = lim A 3 x − 5 3 x − 5 x → ∞ x → ∞ √ . � �� 3 + 1 3 + 1 x 2 � � x 2 � r D x 2 x 2 = lim = lim 3 − 5 3 − 5 � � � � x → ∞ x → ∞ x x x x �� �� 3 + 1 3 + 1 � � | x | x x 2 x 2 = lim = lim 3 − 5 3 − 5 � � � � x → ∞ x x → ∞ x x x √ �� 3 + 1 � 3 x 2 = lim = Dr. Abdulla Eid (University of Bahrain) 3 − 5 � Infinite Limits 11 / 29 � 3 x → ∞

Exercise 7 Find √ 3 x 2 + 1 lim 3 x − 5 x →− ∞ Solution: Direct substitution gives d i 3 ( − ∞ ) 2 + 1 E � ∞ lim = undefined! a 3 ( − ∞ ) − 5 − ∞ x →− ∞ l l u √ √ d 3 x 2 + 1 3 x 2 + 1 b lim = lim A 3 x − 5 3 x − 5 x →− ∞ x → ∞ √ � �� . 3 + 1 3 + 1 x 2 � � x 2 � r D x 2 x 2 = lim = lim 3 − 5 3 − 5 � � � � x →− ∞ x x →− ∞ x x x �� �� 3 + 1 3 + 1 | x | � − x � x 2 x 2 = = lim lim 3 − 5 3 − 5 � � � � x →− ∞ x x →− ∞ x x x √ �� 3 + 1 � − = − 3 x 2 = lim 3 − 5 Dr. Abdulla Eid (University of Bahrain) � Infinite Limits � 12 / 29 3 x →− ∞ x

Multiplying by the conjugate Example 8 Find �� � x 2 + 1 − x lim x → ∞ d i E Solution: Direct substitution gives a �� � l ( ∞ ) 2 + 1 − x l = ∞ − ∞ undefined! u d b A � √ � x 2 + 1 + x . �� � �� � x 2 + 1 − x x 2 + 1 − x r = lim · lim x → ∞ lim � √ D � x → ∞ x → ∞ x 2 + 1 + x x 2 + 1 − x 2 1 = lim � = lim � √ � √ � x → ∞ x 2 + 1 + x x → ∞ x 2 + 1 + x = 0 Dr. Abdulla Eid (University of Bahrain) Infinite Limits 13 / 29

Exercise 9 Find � � � x 2 + 16 lim x − x → ∞ d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Infinite Limits 14 / 29

2 - Horizontal Asymptotes Motivational Example: Consider the function f ( x ) = x 2 − 1 x 2 + 1 . Then we have x → ∞ f ( x ) = 1 lim and x → ∞ f ( x ) = 1 lim d i E In this case, the line y = 1 is called a horizontal asymptote . a l l y u d b A . r D x Dr. Abdulla Eid (University of Bahrain) Infinite Limits 15 / 29

Definition 10 The line y = L is called a horizontal asymptote of the curve y = f ( x ) if either d i E x → ∞ f ( x ) = L lim and x →− ∞ f ( x ) = L lim a l l u Example 11 d b Find the horizontal asymptote of the function A x − 9 . r f ( x ) = √ D 4 x 2 + 3 x + 2 we need to find both lim x → ∞ f ( x ) and lim x →− ∞ f ( x ) Dr. Abdulla Eid (University of Bahrain) Infinite Limits 16 / 29

x − 9 x − 9 lim √ = lim √ 4 x 2 + 3 x + 2 3 x 2 + 3 x + 1 x → ∞ x → ∞ 1 − 9 1 − 9 � � � � x x x x = lim � = lim √ � �� x → ∞ x → ∞ 4 + 3 x + 2 4 + 3 x + 2 x 2 � x 2 � d x 2 x 2 i E 1 − 9 1 − 9 � � � � x x x x = lim � = lim a �� �� x → ∞ x → ∞ 4 + 3 l x + 2 4 + 3 x + 2 � | x | x l u x 2 x 2 d 1 − 9 � � � = 1 b x = lim A �� 2 x → ∞ 4 + 3 x + 2 x 2 . r D Hence y = 1 2 is a horizontal asymptote. Now we compute lim x →− ∞ f ( x ) 2 and so we have y = − 1 to get lim x →− ∞ f ( x ) = − 1 is also a horizontal 2 asymptote. Dr. Abdulla Eid (University of Bahrain) Infinite Limits 17 / 29

Motivation Example Consider the function f ( x ) = 1 x . The graph of the function is y d i E a x l l u d b A . r D Question: What is lim x → 0 + 1 x and lim x → 0 − 1 x ? From the graph we can easily see that � 1 � 1 1 � 1 � x = ∞ 0 + = ∞ x = ∞ 0 + = ∞ lim and lim x → 0 + x → 0 − Dr. Abdulla Eid (University of Bahrain) Infinite Limits 18 / 29

Example 12 Find 3 lim x − 1 d x → 1 + i E Solution: Direct substitution gives a l l 3 u undefined! d 0 b A So we need to find whether it is 0 + or 0 − . . r D x − 1 = 3 3 lim 0 + x → 1 + = ∞ Dr. Abdulla Eid (University of Bahrain) Infinite Limits 19 / 29

Exercise 13 Find 3 lim x − 1 d x → 1 − i E Solution: Direct substitution gives a l l 3 u undefined! d 0 b A So we need to find whether it is 0 + or 0 − . . r D x − 1 = 3 3 lim 0 − x → 1 − = − ∞ Dr. Abdulla Eid (University of Bahrain) Infinite Limits 20 / 29

Example 14 Find − 2 lim x + 1 x →− 1 + d i E Solution: Direct substitution gives a l − 2 l u undefined! 0 d b So we need to find whether it is 0 + or 0 − . A . x + 1 = − 2 − 2 r D lim 0 + x →− 1 + = − 2 · ∞ = − ∞ Dr. Abdulla Eid (University of Bahrain) Infinite Limits 21 / 29

Exercise 15 Find 3 lim 2 − x d x → 2 + i E Solution: Direct substitution gives a l l 3 u undefined! d 0 b A So we need to find whether it is 0 + or 0 − . . r D 2 − x = 3 3 lim 0 − x → 2 + = − ∞ Dr. Abdulla Eid (University of Bahrain) Infinite Limits 22 / 29

Example 16 Find 2 x lim x 2 − 16 d x → 4 − i E Solution: Direct substitution gives a l l 8 u undefined! d 0 b A So we need to find whether it is 0 + or 0 − . . r D x 2 − 16 = 8 2 x lim 0 − x → 4 − = − ∞ Dr. Abdulla Eid (University of Bahrain) Infinite Limits 23 / 29

Example 17 Find x − 2 lim x 2 − 4 x + 4 x → 2 + Solution: Direct substitution gives d i E 0 undefined! a 0 l l u d So we need to factor first using the methods of Section 2.2. b A x − 2 ( x − 2 ) 1 lim x 2 − 4 = lim ( x − 2 )( x − 2 ) = lim . x − 2 x → 2 + r x → 2 + x → 2 + D So we need to find whether it is 0 + or 0 − . x − 2 = 1 1 lim 0 + x → 2 + = ∞ Dr. Abdulla Eid (University of Bahrain) Infinite Limits 24 / 29