3G 4G 10:12 0 0 0.00 2: 41% Document View .. . Senior High School Basic Calculus Quarter 3 - Module 2: Limit Laws ADM ALE NMENT PROPERTY OT FOR S VERN GOVE Basic Calculus - Grade 11 Alternative Delivery Mode Quarter 3- Module 2: Limit Laws First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office m long other things, impose as a con owed mate songs, stories, poems, pictures, ? included in th module are owned by their respecti and seek permission to use the Helpful ight Unhelpful her and authors do not rep Ask Expert Tutors E3G 4G 10:13 0.10 2: 41% Document View .. . 11 1. lim(x2 - 2x - 3) X-5 2. lim x-+2 3. lim (Vx + 15 ) x-+10 4. lim (2x2 - 3x - 4) X -2 5. lim (Vx - 5 ) 6. lim x-3 xz+4x-21 7. lim X--7 X+7 What I Have Learned Express what you have learned in this lesson by answering the questions below. Write your answer on a separate sheet of paper. 1. In what way did the Limit laws help you in solving for the limit of a function? 12 16 E3G 4G 10:13 1.40 B/ s Document View .. . 2. Is direct substitution of x values always applicable in solving the limit of a function? Briefly justify your answer. What I Can Do Tell whether the mathematical statement below is TRUE or FALSE. Explain your answer on a separate sheet of paper. If the limit of two different functions, lim f(x) and lim g(x) exist, then so does lim[f(x) - g(x)]. Assessment Solve for the limit of the following items. Write your answers on a separate sheet of paper. 13 1. lim (10) *+6 A. 11 B. 10 C. 9 D. 8 2. lim(x2 - 3x - 1) X-+5 A. 11 B. 10 179 D. 8 Jim (x2-25\f3G 4G 10:13 0.40 KB/ 6 2: 41x Document View .. . 9. lim [(x - 4) (x - 2)] A. - 3 B. 6 C. - 6 D. 3 10. lim [(2)(x2 + x - 5)] x-+-2 A. - 6 B. O C. - 2 D. 4 11. lim (x3 + x2 - 5x -3)3 X-+2 A. O B. -1 C. 1 D. -2 12. lim (x+2)(x+1)] x-1 (x-1) A. O B. DNE C. 1 D. No answer 13. lim vx + 3 X+-3 A. 1 B. 3 C. O D. 2 14. lim [x(x + 2)(x -2)] A. 16 B. 13 C. 15 D. 12 15. lim 2x-10 X-5 x2-2x-15 A. 2 B. C. 5 D. NI 15 Additional Activities Evaluate the limit of the following items. Write your answer on a separate sheet of paper. 1. lim (Vx-5) x +25 x-25 19 x3+3x2+10x+104\\ E3G 4G 10:13 1.30 2: 40% Document View .. . 15 Additional Activities Evaluate the limit of the following items. Write your answer on a separate sheet of paper. 1. lim (Vx-5) x-25 x-25 lim x3+3x2+10x+104 x2-12 2 " X-4 16 20 References DepEd. 2013. Basic Calculus. Teachers Guide. E3G 4G 10:12 0 Q 0.50 KB/ 4 : 41% Document View .. . Lesson 1 Limit Laws What are laws and why are they created? For sure there will be lots of explanation about it, one of which is that, it is a rule that is meant to be followed for greater good. The Limit lesson has its own laws as well and it was made because of the advantages it can provide in solving the limits of different functions. What's In Solve for the limit of the given item using the table of values. Write your solution on a separate sheet of paper. (Use calculator whenever necessary) Given: lim x-6 X-3 X-3 Table A. (for x values that approaches 3 from the left) X 2.8 2.9 2.99 Table B. (for x values that approaches 3 from the right) X 3.001 3.01 3.1 What's New 8 Read and follow the steps in solving the limit of a function using these different methods. Fill in the bianins to complete solution of the given. Copy and answer the table on a separate3G 4G 10:12 0 Q 0.00 2: 41% Document View .. . What's New Read and follow the steps in solving the limit of a function using these different methods. Fill in the blanks to complete the solution of the given. Copy and answer the table on a separate sheet of paper. Given: lim X-3 Steps Solution 1. Observe the given function. Since it is a rational function, check whether its numerator x2 - x - 6 = (x+_)(x-_ or denominator is factorable. 2. Since the numerator factorable, it is evident that (x - 3) can be divided. lim (x+ 2)(x-3) X-+3 (x-3) 3. What is left is just (x + 2), since it is a polynomial function; direct substitution is lim (x + 2) applicable because it has no X-+3 domain restrictions. 4. Perform the operation. [(3) + 2] =. 5. Indicate the final answer. lim - x - 6 x-+3 x - 3 5 What is It 9 LIMIT laws ar sed as alternative ways in solving the limit of a function3G 4G 10:12 0 Q 0.00 41x Document View .. . What is It Limit laws are used as alternative ways in solving the limit of a function without using table of values and graphs. Below are the different laws that can be applied in various situations to solve for the limit of a function. A. The limit of a constant is itself. If k is any constant, then, lim (k) = k x-c Example: 1. lim(5) = 5 2. lim(-9) = -9 x-c B. The limit of x as x approaches c is equal to c. That is, lim x = C Examples: 1. lim (x) = 8 2. lim (x) = -2 X--2 ? For the remaining theorems, we will assume that the limits of f and g both exist as x approaches c and that they are L and M, respectively. In other words, lim f(x) = L and lim g(x) = M C. The Constant Multiple Theorem. The limit of a constant & times a function is equal to the product of that constant and its function's limit [k . f(x)] = k . lim f(x) = k . L 6 Examples: If lim f(x) = 3 , then 1. lim 5 . f(x) = 5 . lim f(x) = 103 = 15 " lim (-9) f(x) = (-9) iim fly) = (-9) 3= -273G 4G 10:12 0 Q 0.30 KB/ 4 2: 41% Document View .. . Examples: If lim f(x) = 3 , then 1. lim 5. f(x) = 5 . lim f(x) = 5 . 3 = 15 x->c 2. lim (-9) . f(x) = (-9) . lim f(x) = (-9) . 3= -27 D. The Addition theorem. The limit of a sum of functions is the sum of the limits of the individual functions. lim [ f(x) + g(x) ] = lim f(x) + lim g(x) = L + M x c Examples: 1. If lim f(x) = 3 and lim g(x) = -4, then lim ( f(x) + g(x)) = lim f(x) + lim g(x) = 3 + (-4) = -1 2. If lim f(x) = -5 and lim g(x) = -2, then lim( f(x) + g(x)) = lim f(x) + lim g(x) = -5+ (-4) = -9 x-C E. The Subtraction Theorem. The limit of a difference of functions is the difference of the limits of the individual functions. lim [ f(x) - g(x)] = lim f(x) - lim g(x) = L - M x-+c Examples: 1. If lim f(x) = 3 and lim g(x) = -4, then lim ( f(x) - g(x)) = lim f(x) - lim g(x) = 3 - (-4) = 7 2. If lim f(x) = -5 and lim g(x) = -2, then 7 lim ( f(x) - g(x)) = lim f(x) - lim g(x) = -5- (-4) = -1 F. The Multiplication Theorem. The limit of a product of functions is the product of the limits of the individual functions. lim [ f(x) . e(x) = um f(x) . lim g(x) = L . M3G 4G 10:12 0 Q 0.00 2: 41% Document View .. . V lim ( f(x) - g(x)) = lim f(x) - lim g(x) = -5 - (-4) = -1 F. The Multiplication Theorem. The limit of a product of functions is the product of the limits of the individual functions. lim [ f(x) . g(x)] = lim f(x) . lim g(x) = L . M Examples: 1. If lim f(x) = 3 and lim g(x) = -4, then lim( f(x) . g(x)) = lim f(x) . lim g(x) = (3)(-4) = -12 2. If lim f(x) = -5 and lim g(x) = -2, then lim ( f(x) . g(x)) = lim f(x) . lim g(x) = (-5)(-4) = 20 G. The Division Theorem. The limit of a quotient of functions is the quotient of the limits of the individual functions, provided that the denominator is not equal to zero. lim f(x) lim [f (x) X-C x-c g(x) M#0 lim g(x) M Examples: 1. If lim f (x) = 3 and lim g(x) = -6, then x-c x-c lim f (x) lim f (x) x+c 3 x-clg(x) lim g (x) -6 X-c 2. If lim f (x) = 0 and lim g(x) = 7, then X-c x-c f(x) lim f (x) 0 lim x-c g(x) lim g (x) x-c 8 H. The Power Theorem. The limit of an integer power p of a function is just that power of the limit of the function. lim [f(x )]P 12im f(x)] = (LY x->c Examples:3G 4G 10:13 0.40 KB/ 6 2: 41x Document View .. . H. The Power Theorem. The limit of an integer power p of a function is just that power of the limit of the function. lim [f(x )]P = [lim f(x)] = (Ly Examples: 1. If lim f(x) = 3, then lim[f(x)]* = [lim f(x)] = (3)4 = 81 2. If lim f(x) = -4, then lim[.f(x)]3= [lim f(x)] = (-4)3= -64 x-c 1. The Radical/Root Theorem. If n is a positive integer, the limit of the nik root of a function is just the at root of the limit of the function, provided that the nik root of the limit is a real number. lim \\f(x) = " (lim f(x) = VI Examples: X-C 1. If lim f (x) = 8, then x-c lim Vf (x) = 3 (lim f (x) = V8 = 2 x-c 2. If lim f (x) = 64, then X-+C lim Vf (x) = ( lim f (x) = V64 = 8 x-+c J 9 More examples: 1. Find: lim (x* + 4x - 3) x-4 13 Solution: Steps Solution3G 4G 10:13 1.00 B/ & 2. 41% Document View .. . 9 More examples: 1. Find: lim (x2 + 4x - 3) X-4 Solution: Steps Solution 1. Apply Addition Law Theorem. lim (x?) + lim (4x) + lim(-3) x++4 x-+4 x-+4 2. Apply Power Theorem on the first term. lim x "+ lim (4x) + lim (-3) 3. Apply Multiplication Theorem on the second term. [Lim x) + 4 [lim x] + lim(-3) 4. Apply the limit of x as 42 + 4(4) + lim (-3) x approaches c is equal to c. X-+4 5. Apply the limit of a constant is the constant itself. 42 + 4(4) + (-3) 6. Simplify. 16 + 16 - 3 = 29 3. Determine: lim (v66 - x ) Solution: Steps Solution 1. Apply Radical/Root Theorem. Tim (66 - x) 2. Apply Subtraction Theorem. lim 66 - lim x VX-+2 X-+2 3. Apply the limit of a constant is the constant itself and the limit of x as x approaches c is V66 - 2 equal to c. 4. Simplify. V64 = 8 10 14 4. Evaluate: lim x-33G 4G 10:13 0.40 B/ & Document View .. . 4. Evaluate: lim X-3 2-x-6 Solution Steps Solution 1. Factor the denominator then lim (x-3) -=lim simplify. x-3 (x+2)(x-3) X-+3L lim (1) 2. Apply Division Theorem. lim (x + 2) 3. Apply Addition Theorem on lim (1) the denominator. lim (x) + lim(2) X-+3 X-+3 4. Apply the limit of a constant is the constant itself and the limit of x as x approaches 3 +2 c is equal to c. 5. Simplify. What's More Determine the limits of the following items using the limit laws. Write your complete solutions on a separate sheet of paper. 11 1. lim(x2 - 2x - 3) 2. lim x-+2 3. lim (Vx + 15 ) x-10 15 4. lim (2x - 3x - 4)