Calculus I - Chapter 1 - Limits and Continuity PDF

Friendly Reminder from the Mathematics Department: Stay healthy! Follow the DOH Health Protocols!. GOD BLESS…..... Math 13 | Calculus I Chapter 1. Limits and Continuity  Chapter 1. Limits and Continuity...

Friendly Reminder from the Mathematics Department: Stay healthy! Follow the DOH Health Protocols!. GOD BLESS…..... Math 13 | Calculus I Chapter 1. Limits and Continuity  Chapter 1. Limits and Continuity Definition of a Limit LIMIT Limit Theorems OF A One –Sided Limits FUNCTION Infinite Limits. Limits at Infinity.. Continuity of a Function... LIMIT OF A FUNCTION How does f(x) behave?  How does 𝑓 𝑥 behave? Behaviour of the function 𝒇(𝒙) Trend of the values Graph of 𝑓(𝑥)......  How does 𝑓 𝑥 behave? Example. Complete the table below using the function 𝑓(𝑥) for the specified values of 𝑥. 𝑥2 − 4 𝑓 𝑥 = 𝑓 𝑥 = 𝑥 2 −4 = (𝑥+2)(𝑥−2) = 𝑥 + 2, 𝑥 ≠ 2 𝑥−2 𝑥−2 𝑥−2 x 0 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.99 1.999 1.9999 1.99999 1.999999 y. x 2.000001 2.00001 2.0001 2.001 2.01 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 4. y....  𝑥2 − 4 𝑓 𝑥 = How does 𝑓 𝑥 behave? 𝑥−2 x 0 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.99 1.999 1.9999 1.99999 1.999999 y 2 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.99 3.999 3.9999 3.99999 3.999999 y 7 One-sided limits Limit of 𝑓(𝑥) as 𝑥 approaches 4 6 2 from the right is 4. 2+ 5 4 Limit of 𝑓(𝑥) as 𝑥 approaches 2 is 4. Two-sided Limit 3 2 Limit of 𝑓(𝑥) as 𝑥 approaches 2 from the left is 4. 2− 4 4. 1 2 0. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x.. x 2.000001 2.00001 2.0001 2.001 2.01 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 4 y 4.000001 4.00001 4.0001 4.001 4.01 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 6..  Informal Definition of a Limit Consider a function 𝑓 of a single variable 𝑥. Consider a constant 𝑎 which the variable 𝑥 approach (𝑎 may or may not be in the domain of 𝑓). The limit, denoted by 𝐿, is the unique real value that 𝑓(𝑥) will approach as 𝑥 approaches 𝑎. In symbols, we write this process as.. 𝑎....  y 7 4 6 2 5 L 4 3 2 1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 𝐿 + 𝜀 = 𝒇(𝒙𝟐 ) L 𝜀. 𝜀 𝛿1 𝛿2. 𝐿 − 𝜀 = 𝒇(𝒙𝟏 ) 𝒙𝟏 𝒂 𝒙𝟐 𝛿 = min{𝛿1 , 𝛿2 }....  Formal Definition of a Limit...... LIMIT THEOREMS (Part 1) What are the Limit properties?  Limit Theorems 𝑓 𝑥 =4. lim 4 = 4 lim 4 = 4 lim 4 = 4. 𝑥→2 𝑥→0 𝑥→3.. How about 𝑥 → −5??? lim 4 = 4. 𝑥→−5.  Limit Theorems 𝑓 𝑥 =𝑥. lim 𝑥 = 2 lim 𝑥 = 0 lim 𝑥 = 3. 𝑥→2 𝑥→0 𝑥→3.. How about 𝑥 → −5??? lim 𝑥 = −5. 𝑥→−5.  Limit Theorems lim 𝑥 + 7 = lim 𝑥 + lim 7 = 4 + 7 = 11 𝑥→4 𝑥→4 𝑥→4... lim 2𝑥 − 3 + 3𝑥 − 5 = lim 2𝑥 − 3 +lim 3𝑥 − 5 𝑥→4 𝑥→4 𝑥→4...  Limit Theorems lim 2𝑥= lim 2 lim 𝑥 = 2 3 = 6 𝑥→3 𝑥→3 𝑥→3.. lim 2𝑥 3𝑥 − 1 = lim 2𝑥 lim (3𝑥 − 1). 𝑥→4 𝑥→4 𝑥→4...  Polynomial Function 𝑓 𝑥 = 𝑐0 + 𝑐1 𝑥 + 𝑐2 𝑥 2 + ⋯ + 𝑐𝑛 𝑥 𝑛 lim 𝑓 𝑥 = lim 𝑐0 + 𝑐1 𝑥 + 𝑐2 𝑥 2 + ⋯ + 𝑐𝑛 𝑥 𝑛 = 𝑓(𝑎) 𝑥→𝑎 𝑥→𝑎... Remark. Substitute 𝑎 in 𝑓(𝑥) to obtain the limit of a polynomial. function as 𝑥 approaches 𝑎...  Evaluate the following: 1. lim 2𝑥 3 + 3𝑥 2 − 1 = 2(23 ) + 3(22 ) − 1 = 27 𝑥→2 2. lim 3𝑥 7 − 5𝑥 4 + 2𝑥 3 + 9 = −1 𝑥→−1......  Summary Limit Theorems Polynomial Function... Substitute 𝑎 in 𝑓(𝑥) to obtain the limit of the polynomial. function as x approaches 𝑎... LIMIT THEOREMS (Part 2) What are the Limit properties?  Polynomial Functions Substitution Rational Functions 𝑓(𝑥) ℎ 𝑥 = ,𝑔 𝑥 ≠ 0 𝑔(𝑥) 𝑛𝑜𝑛𝑧𝑒𝑟𝑜/𝑧𝑒𝑟𝑜 𝑧𝑒𝑟𝑜 𝑛𝑜𝑛𝑧𝑒𝑟𝑜 ℎ 𝑎 = ℎ 𝑎 = ℎ 𝑎 = 𝑛𝑜𝑛𝑧𝑒𝑟𝑜 𝑧𝑒𝑟𝑜 𝑧𝑒𝑟𝑜 Reduced to a You need to apply Limit Does Not Exist. real number something Infinite Limits.. What to do first? Try substituting the value of 𝑎 in the given rational function and check which case it falls into....  Limit Theorems 𝑛𝑜𝑛𝑧𝑒𝑟𝑜/𝑧𝑒𝑟𝑜 ℎ 𝑎 = lim (2𝑥 − 1) 1 2𝑥 − 1 𝑥→1 𝑛𝑜𝑛𝑧𝑒𝑟𝑜 lim = = 2𝑥 − 1 2(1) − 1 1 𝑥→1 3𝑥 + 5 lim (3𝑥 + 5) 8 = = 𝑥→1 3𝑥 + 5 3(1) + 5 8.. 𝑥2 + 𝑥 − 2 lim (𝑥 2 + 𝑥 − 2) 4 𝑥2 + 𝑥 − 2 (2)2 +(2) − 2 4 lim 𝑥→2 = =. 𝑥→2 2𝑥 2 − 𝑥 − 1 = 2 = 2𝑥 2 − 𝑥 − 1 2(22 ) − (2) − 1 5 lim (2𝑥 − 𝑥 − 1) 5. 𝑥→2..  𝑥2 + 𝑥 − 2 4 Limit Theorems lim 2 = 𝑧𝑒𝑟𝑜 𝑥→2 2𝑥 − 𝑥 − 1 5 ℎ 𝑎 = 𝑧𝑒𝑟𝑜 𝑥2 + 𝑥 − 2 lim 2 =? ? ? 𝑥→1 2𝑥 − 𝑥 − 1 𝑥2 + 𝑥 − 2 (1)2 +(1) − 2 0 = = 2𝑥 2 − 𝑥 − 1 2(12 ) − (1) − 1 0 Do something! 𝑥2 + 𝑥 − 2 (𝑥 − 1)(𝑥 + 2) lim 2 = lim 𝑥→1 2𝑥 − 𝑥 − 1 𝑥→1 (𝑥 − 1)(2𝑥 + 1). 𝑥+2 lim (𝑥 + 2) 3 𝑥→1. = lim = = =1 𝑥→1 2𝑥 + 1 lim (2𝑥 + 1) 3. 𝑥→1...  Limit Theorems 𝑥+2−2 𝑧𝑒𝑟𝑜 lim =? ? ? ℎ 𝑎 = 𝑥→2 𝑥−2 𝑧𝑒𝑟𝑜 𝑥+2−2 2+2−2 0 = = Do something! 𝑥−2 2−2 0 𝑥+2−2 𝑥+2−2 𝑥+2+2 lim = lim ∙ 𝑥→2 𝑥−2 𝑥→2 𝑥−2 𝑥+2+2. (𝑥 + 2) − 4. = lim. 𝑥→2 (𝑥 − 2)( 𝑥 + 2 + 2). 𝑥−2 1 1 = lim = lim =. 𝑥→2 (𝑥 − 2)( 𝑥 + 2 + 2) 𝑥→2 𝑥 + 2 + 2) 4.  Limit Theorems 𝑥2 + 1 𝑛𝑜𝑛𝑧𝑒𝑟𝑜 lim =? ? ? ℎ 𝑎 = 𝑥→2 𝑥 − 2 𝑧𝑒𝑟𝑜 𝑥 2 + 1 22 + 1 5 The Limit does not exist. 𝑥−2 = 2−2 = 0 𝑥2 + 1 lim does not exist 𝑥→2 𝑥 − 2. The behaviour of the graph for this case can be. observed using the concept of infinite limits.....  Limit Theorems Existence of Limits The limit exists if we find a single real function value upon applying the limit theorems. Uniqueness of Limits If lim 𝑓 𝑥 = 𝐿1 and lim 𝑓 𝑥 = 𝐿2 , then 𝐿1 = 𝐿2. 𝑥→𝑎 𝑥→𝑎......  Summary Rational Functions 𝑓(𝑥) ℎ 𝑥 = ,𝑔 𝑥 ≠ 0 𝑔(𝑥) 𝑛𝑜𝑛𝑧𝑒𝑟𝑜/𝑧𝑒𝑟𝑜 𝑧𝑒𝑟𝑜 𝑛𝑜𝑛𝑧𝑒𝑟𝑜 ℎ 𝑎 = ℎ 𝑎 = ℎ 𝑎 = 𝑛𝑜𝑛𝑧𝑒𝑟𝑜 𝑧𝑒𝑟𝑜 𝑧𝑒𝑟𝑜 Reduced to a You need to apply Limit Does Not Exist. real number something Infinite Limits..... LIMIT THEOREMS (Part 3) What are the Limit properties?  Limit Theorems Radical functions 𝑛 ℎ(𝑥) = 𝑓(𝑥)......  Limit Theorems 𝑓 𝑥 = 𝑥 lim 𝑥 = 2 Why? 𝑥→4 lim 𝑥 = lim 𝑥 = 4 = 2 𝑥→4 𝑥→4... How about lim 𝑥 ??? lim 𝑥 = 3. 𝑥→9 𝑥→9..  Limit Theorems lim 𝑥 =? ? ? 𝑥→−4 lim 𝑥 does not exist Why? 𝑥→−4 lim 𝑥 = lim 𝑥 = −4 𝑥→−4 𝑥→−4 Which is not a real number.. Try this!. lim 2𝑥 𝑥 − 6 + 18.. 𝑥→6.  Limit Theorems lim 2𝑥 𝑥 − 6 + 18 = lim [2𝑥 𝑥 − 6 + 18] 𝑥→6 𝑥→6 = 18 = 2(9) =3 2.. Try this!. 3. lim 𝑥 2 2𝑥 2 − 5 + 2𝑥 − 7. 𝑥→2.  Limit Theorems 3 lim 𝑥 2 2𝑥 2 − 5 + 2𝑥 − 7 𝑥→2 3 = lim [𝑥 2 2𝑥 2 − 5 + 2𝑥 − 7] 𝑥→2 3 = 9... 3𝑥 − 7 Try this! lim 𝑥→1 2𝑥 2 − 9𝑥 − 5...  Limit Theorems 3𝑥 − 7 3𝑥 − 7 lim = lim 𝑥→1 2𝑥 2 − 9𝑥 − 5 𝑥→1 2𝑥 2 − 9𝑥 − 5 lim [3𝑥 − 7] 𝑥→1 = lim [2𝑥 2 − 9𝑥 − 5] 𝑥→1.. −4 1 3. = = =. −12 3 3..  The limit does not exist when 𝑛. 𝐿 is not a real number......