Lesson 1 - Limit of Function PDF

Who is absent today?  Classroom Rules 1. Raise your hand if you want to answer. 2. No using of cellphones. 3. Respect each other. LIMIT OF A FUNCTION LIMIT are the backbone of calculus, and ca...

Who is absent today?  Classroom Rules 1. Raise your hand if you want to answer. 2. No using of cellphones. 3. Respect each other. LIMIT OF A FUNCTION LIMIT are the backbone of calculus, and calculus is called the Mathematics of Change. The study of limits is necessary in studying change in great detail. The evaluation of a particular limit is what underlies the formulation of the derivative and the integral of a function.  This is how it is with limits of functions. We will consider functions of a single variable and study the behavior of functions as its variable approaches a particular value (a constant).  The variable can only take values very, very close to the constant, but it cannot equal the constant itself. However, the limit will be able to describe clearly what is happening to the function near the constant. DEFINITION OF LIMIT If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, then the limit of f(x) as x approaches C is L. In symbols, we write this process as lim f(x) = L x→c This is read, “ The limit of f(x) as x approaches c is L.” Example: lim 3x² lim 2x / x²+1 x→4 x→ 0 𝟑 lim 𝒙 x→∞  ESTIMATING LIMIT NUMERICALLY Example 1: lim (3x -2) x →2 X 1.9 1.99 1.999 2 2.001 2.01 F(X) 3.7 3.97 3.997 ? 4.003 4.03 Therefore, lim (3x-2) = 4 x →2  𝟐 Example 2: lim - x x →3 𝟑 Substitution method: Let x = 3 2 = 3-x Therefore, The limit does NOT exist. 2 = 3-3 2 = 0 = undefined  𝒙𝟐 −𝟗 Example 3: lim 𝒙 −𝟑 Substitution method: Let x = 3 = x² - 9 x-3 3² - 9 = 3-3 9-9 = 3-3 0 = 0 = indeterminate By Factoring Technique: By Substitution lim (x+3) x² - 9 x→3 = x-3 Let x = 3 = (x-3) (x+3) = x+3 x-3 = 3+3 = x+3 = 6 Therefore, The limit of the function exists. Example 4: Find the lim (x+2) x→4 X 2 2.5 3 3.1 3.3 3.5 3.7 3.9 3.99 3.999 F(x) 4 4.5 5 5.1 5.3 5.5 5.7 5.9 5.99 5.999 Example 4: Find the lim (x+2) x→4 Table 2 X 6 5.5 5 4.9 4.7 4.5 4.3 4.1 4.01 4.001 F(x) 8 7.5 7 6.9 6.7 6.5 6.3 6.1 6.01 6.001  lim (x +2) =6 x →4 Therefore, The limit of the function exists. By Substitution method lim (x +2) x →4 =x+2 =4+2 =6 Solve the following: 1. lim (2x + 1) x→2 2. lim (2x - 6) x→4 𝒙𝟐 −𝟒 3. lim 𝒙 −𝟐 x→3  ANY QUESTIONS? Thank You Everyone!