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Basic Calculus Lesson 3

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Limits of Non - Algebraic Function - Limit of an Exponential Function - Limit of Logarithmic Function

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Limit of an Exponential Function Let a and b be real numbers, where b > 0 and b ≠ 1. Then, 𝐥𝐢𝐦 𝒙→𝒂 𝒃 𝒙 = 𝒃 𝒂 .
Properties There are four basic properties in limits, which are used as formulas in evaluating the limits of exponential functions. 1.Power Rule 2.Constant Base Power Rule 3.Constant Exponent Power Rule 4.Radical Power Rule
1. Power Rule 𝒍𝒊𝒎 𝒙→𝒂 𝒇 𝒙 𝒈 𝒙 = 𝒍𝒊𝒎 𝒙→𝒂 𝒇 𝒙 𝒍𝒊𝒎 𝒙→𝒂 𝒈 𝒙 It is a property of power rule, used to find the limit of an exponential function whose base and exponent are in a function form.
1. Power Rule 𝒍𝒊𝒎 𝒙→𝒂 𝒇 𝒙 𝒈 𝒙 = 𝒍𝒊𝒎 𝒙→𝒂 𝒇 𝒙 𝒍𝒊𝒎 𝒙→𝒂 𝒈 𝒙 Example: Evaluate 𝒇 𝒙 = 𝒙 𝟐 − 𝟒, 𝒈 𝒙 = 𝟐𝒙 ; 𝐥𝐢𝐦 𝒙→𝟑 𝒇 𝒙 𝒈 𝒙 Solution: 𝐥𝐢𝐦 𝒙→3 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→3 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝟑 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→3 𝒇 𝒙2 − 4 𝐥𝐢𝐦 𝒙→3 𝒈 2𝒙 = 𝟑 𝟐 − 𝟒 𝟐 𝟑 = 𝟗 − 𝟒 𝟔 = 𝟓 𝟔 = 𝟏𝟓, 𝟔𝟐𝟓
2. Constant Base Power Rule 𝒍𝒊𝒎 𝒙→𝒂 𝒃 𝒇 𝒙 = 𝒍𝒊𝒎 𝒙→𝒂 𝒃 𝒍𝒊𝒎 𝒙→𝒂 𝒇 𝒙 The limit of an exponential function is equal to the limit of the exponent with same base. It is called the limit rule of an exponential function.
2. Constant Base Power Rule 𝒍𝒊𝒎 𝒙→𝒂 𝒃 𝒇 𝒙 = 𝒍𝒊𝒎 𝒙→𝒂 𝒃 𝒍𝒊𝒎 𝒙→𝒂 𝒇 𝒙 Example: Solution: Evaluate 𝑏 = 12, 𝑓 𝑥 = 𝑥 + 3; 𝐥𝐢𝐦 𝒙→𝟏 𝒃 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→𝟏 𝒃 )𝒇(𝒙 = 𝒍𝒊𝒎 𝒙→𝟏 𝟏𝟐 𝒙+𝟑 = 𝒍𝒊𝒎 𝒙→𝟏 𝟏𝟐 𝒍𝒊𝒎 𝒙→𝟏 𝒙+𝟑 = 𝟏𝟐 𝒙+𝟑 = 𝟏𝟐 𝟏+𝟑 = 𝟏𝟐 𝟒 = 𝟐𝟎, 𝟕𝟑𝟔
3. Constant Exponent Power Rule 𝒍𝒊𝒎 𝒙→𝒂 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→𝒂 𝒇 𝒙 𝒏 Example: Solution: Evaluate 𝑓 𝑥 = 16𝑥2 − 64, 𝑛 = 2; 𝐥𝐢𝐦 𝒙→𝟏 𝒇 𝒙 𝒏 𝒍𝒊𝒎 𝒙→𝟏 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→𝟏 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→𝟏 𝟏𝟔𝒙 𝟐 − 𝟔𝟒 𝟐 = 𝟏𝟔 𝟏 𝟐 − 𝟔𝟒 𝟐 = 𝟏𝟔 − 𝟔𝟒 𝟐 = −𝟒𝟖 𝟐 = 𝟐, 𝟑𝟎𝟒
4. Radical Power Rule 𝒍𝒊𝒎 𝒙→𝒂 𝒏 )𝒇(𝒙 = 𝒏 𝒍𝒊𝒎 𝒙→𝒂 )𝒇(𝒙 Example: Solution: Evaluate 𝒇 𝒙 = 𝒙 𝟐 − 𝟔𝒙 + 𝟗, 𝒏 = 𝟐; 𝒍𝒊𝒎 𝒙→𝟔 𝒏 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→𝟔 𝒏 )𝒇(𝒙 = 𝒏 𝒍𝒊𝒎 𝒙→𝟔 )𝒇(𝒙 = 𝐥𝐢𝐦 𝒙→6 𝒙2 − 6𝒙 + 9 = 𝒙 − 3 2 = 𝒙 − 3 = 6 − 3 = 𝟑
Let’s Practice!!! 1. Evaluate 𝒇 𝒙 = 𝟒 𝒙 𝟐 − 𝟐𝟒𝒙 + 𝟑𝟔, 𝒏 = 𝟐; a. 𝒍𝒊𝒎 𝒙→−𝟑 𝒏 𝒇(𝒙) b. 𝒍𝒊𝒎 𝒙→𝟒 𝒇 𝒙 𝒏 2. Evaluate 𝒇 𝒙 = 𝒙 𝟑 − 𝟐𝟕, 𝒈 𝒙 = 𝟐𝒙 + 𝟏 a. 𝒍𝒊𝒎 𝒙→𝟐 𝒇 𝒙 𝒈 𝒙 3. Evaluate 𝑏 = 𝟑, 𝑓 𝑥 = 𝒙 − 𝟒 a. 𝐥𝐢𝐦 𝒙→𝟖 𝒃 𝒇(𝒙)
1. Evaluate 𝒇 𝒙 = 𝟒 𝒙 𝟐 − 𝟐𝟒𝒙 + 𝟑𝟔, 𝒏 = 𝟐; a. 𝒍𝒊𝒎 𝒙→−𝟑 𝒏 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→−𝟑 𝒏 )𝒇(𝒙 = 𝒏 𝒍𝒊𝒎 𝒙→−𝟑 )𝒇(𝒙 = 𝐥𝐢𝐦 𝒙→−𝟑 𝟒𝒙2 − 𝟐𝟒𝒙 + 𝟑𝟔 = 𝟐𝒙 − 𝟔 2 = 𝟐𝒙 − 𝟔 = 𝟐 −𝟑 − 𝟔 = −𝟏𝟐 = −𝟔 − 𝟔
1. Evaluate 𝒇 𝒙 = 𝟒 𝒙 𝟐 − 𝟐𝟒𝒙 + 𝟑𝟔, 𝒏 = 𝟐; b. 𝒍𝒊𝒎 𝒙→𝟒 𝒇 𝒙 𝒏 𝒍𝒊𝒎 𝒙→𝟒 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→𝟒 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→𝟒 𝟒𝒙 𝟐 − 𝟐𝟒𝒙 + 𝟑𝟔 𝟐 = 𝟒 𝟒 𝟐 − 𝟐𝟒 𝟒 + 𝟑𝟔 𝟐 = 𝟒 𝟏𝟔 − 𝟗𝟔 + 𝟑𝟔 𝟐 = 𝟒 𝟐 = 𝟏𝟔 = 𝟔𝟒 − 𝟗𝟔 + 𝟑𝟔 𝟐
2. Evaluate 𝒇 𝒙 = 𝒙 𝟑 − 𝟐𝟕, 𝒈 𝒙 = 𝟐𝒙 + 𝟏 a. 𝒍𝒊𝒎 𝒙→𝟐 𝒇 𝒙 𝒈 𝒙 𝐥𝐢𝐦 𝒙→𝟐 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→2 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝟐 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝟐 𝒙 𝟑 − 𝟐𝟕 𝐥𝐢𝐦 𝒙→𝟐 𝟐𝐱+𝟏 = 𝟐 𝟑 − 𝟐𝟕 𝟐 𝟐 +𝟏 = 𝟖 − 𝟐𝟕 𝟓 = −𝟏𝟗 𝟓 = −𝟐, 𝟒𝟕𝟔, 𝟎𝟗𝟗
3. Evaluate 𝑏 = 𝟑, 𝑓 𝑥 = 𝒙 − 𝟒 a. 𝐥𝐢𝐦 𝒙→𝟖 𝒃 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→𝟖 𝒃 )𝒇(𝒙 = 𝒍𝒊𝒎 𝒙→𝟖 𝟑 𝒙−𝟒 = 𝒍𝒊𝒎 𝒙→𝟖 𝟑 𝒍𝒊𝒎 𝒙→𝟖 𝒙−𝟒 = 𝟑 𝒙−𝟒 = 𝟑 𝟖−𝟒 = 𝟑 𝟒 = 𝟖𝟏
Seatwork 3.1
1. Evaluate 𝒇 𝒙 = 𝒙 𝟐 + 𝟏𝟔𝒙 + 𝟖, 𝒏 = 𝟑; a. 𝒍𝒊𝒎 𝒙→𝟒 𝒏 𝒇(𝒙) b. 𝒍𝒊𝒎 𝒙→−𝟐 𝒇 𝒙 𝒏 2. Evaluate 𝒇 𝒙 = 𝒙 𝟐 − 𝟔𝟒, 𝒈 𝒙 = 𝒙 − 𝟓 a. 𝒍𝒊𝒎 𝒙→𝟗 𝒇 𝒙 𝒈 𝒙 b. 𝒍𝒊𝒎 𝒙→𝟗 𝒈 𝒙 𝒇 𝒙 3. Evaluate 𝑏 = 𝟏𝟓, 𝑓 𝑥 = 𝒙 𝟐 + 𝟐𝒙 + 𝟏 a. 𝐥𝐢𝐦 𝒙→𝟐 𝒃 𝒇(𝒙) b. 𝐥𝐢𝐦 𝒙→−𝟑 𝒃 𝒇(𝒙)
ANSWERS
1. Evaluate 𝒇 𝒙 = 𝒙 𝟐 + 𝟏𝟔𝒙 + 𝟖, 𝒏 = 𝟑; a. 𝒍𝒊𝒎 𝒙→𝟒 𝒏 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→𝟒 𝒏 )𝒇(𝒙 = 𝒏 𝒍𝒊𝒎 𝒙→𝟒 )𝒇(𝒙 = 𝟑 𝒍𝒊𝒎 𝒙→𝟒 𝒙 𝟐 + 𝟏𝟔𝒙 + 𝟖 = 3 𝟒 𝟐 + 𝟏𝟔 𝟒 + 𝟖 = 3 𝟏𝟔 + 𝟔𝟒 + 𝟖 = 𝟒. 𝟒𝟓 = 3 𝟖𝟖
1. Evaluate 𝒇 𝒙 = 𝒙 𝟐 + 𝟏𝟔𝒙 + 𝟖, 𝒏 = 𝟑; b. 𝒍𝒊𝒎 𝒙→−𝟐 𝒇 𝒙 𝒏 𝒍𝒊𝒎 𝒙→−𝟐 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→−𝟐 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→−𝟐 𝒙 𝟐 + 𝟏𝟔𝒙 + 𝟖 𝟑 = −𝟐 𝟐 + 𝟏𝟔 −𝟐 + 𝟖 𝟑 = 𝟏𝟔 − 𝟑𝟐 + 𝟖 𝟑 = −𝟖 𝟑 = −𝟓𝟏𝟐
2. Evaluate 𝒇 𝒙 = 𝒙 𝟐 − 𝟔𝟒, 𝒈 𝒙 = 𝒙 − 𝟓 a. 𝒍𝒊𝒎 𝒙→𝟗 𝒇 𝒙 𝒈 𝒙 𝐥𝐢𝐦 𝒙→𝟗 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→9 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝟗 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝟗 𝒙 𝟐 − 𝟔𝟒 𝐥𝐢𝐦 𝒙→𝟗 𝐱−𝟓 = 𝟗 𝟐 − 𝟔𝟒 𝟗−𝟓 = 𝟖𝟏 − 𝟔𝟒 𝟒 = 𝟏𝟕 𝟒 = 𝟖𝟑, 𝟓𝟐𝟏
2. Evaluate 𝒇 𝒙 = 𝒙 𝟐 − 𝟔𝟒, 𝒈 𝒙 = 𝒙 − 𝟓 b. 𝒍𝒊𝒎 𝒙→𝟗 𝒈 𝒙 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝟗 𝒈 𝒙 𝒇 𝒙 = 𝐥𝐢𝐦 𝒙→9 𝒈 𝒙 𝐥𝐢𝐦 𝒙→𝟗 𝒇 𝒙 = 𝐥𝐢𝐦 𝒙→𝟗 𝒙 − 𝟓 𝐥𝐢𝐦 𝒙→𝟗 𝐱 𝟐−𝟔𝟒 = (𝟗 − 𝟓) 𝟗 𝟐−𝟔𝟒 = (𝟒) 𝟖𝟏−𝟔𝟒 = 𝟒 𝟏𝟕 = 𝟏𝟕, 𝟏𝟕𝟗, 𝟖𝟔𝟗, 𝟏𝟖𝟒
3. Evaluate 𝑏 = 𝟏𝟓, 𝑓 𝑥 = 𝒙 𝟐 + 𝟐𝒙 + 𝟏 a. 𝐥𝐢𝐦 𝒙→𝟐 𝒃 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→𝟐 𝒃 )𝒇(𝒙 = 𝒍𝒊𝒎 𝒙→𝟐 𝟏𝟓 𝒙 𝟐+𝟐𝒙+𝟏 = 𝒍𝒊𝒎 𝒙→𝟐 𝟏𝟓 𝒍𝒊𝒎 𝒙→𝟐 𝒙 𝟐+𝟐𝒙+𝟏 = 𝟏𝟓 𝒙 𝟐+𝟐𝒙+𝟏 = 𝟏𝟓 𝟐 𝟐+𝟐 𝟐 +𝟏 = 𝟏𝟓 𝟗 = 𝟑𝟖, 𝟒𝟒𝟑, 𝟑𝟓𝟗, 𝟑𝟕𝟓
3. Evaluate 𝑏 = 𝟏𝟓, 𝑓 𝑥 = 𝒙 𝟐 + 𝟐𝒙 + 𝟏 b. 𝐥𝐢𝐦 𝒙→−𝟑 𝒃 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→−𝟑 𝒃 )𝒇(𝒙 = 𝒍𝒊𝒎 𝒙→−𝟑 𝟏𝟓 𝒙 𝟐+𝟐𝒙+𝟏 = 𝒍𝒊𝒎 𝒙→−𝟑 𝟏𝟓 𝒍𝒊𝒎 𝒙→−𝟑 𝒙 𝟐+𝟐𝒙+𝟏 = 𝟏𝟓 𝒙 𝟐+𝟐𝒙+𝟏 = 𝟏𝟓 −𝟑 𝟐+𝟐 −𝟑 +𝟏 = 𝟏𝟓 𝟒 = 𝟓𝟎, 𝟔𝟐𝟓
Quiz 3.1
Evaluate 𝒇 𝒙 = 𝟏𝟔𝒙 𝟐 − 𝟖𝒙 + 𝟏, 𝒈 𝒙 = 𝟔𝟒𝒙 𝟐 + 𝟒𝟖𝒙 + 𝟗, 𝒏 = 𝟐; a. 𝒍𝒊𝒎 𝒙→𝟓 𝒏 𝒇(𝒙) b. 𝒍𝒊𝒎 𝒙→−𝟓 𝒇 𝒙 𝒏 c. 𝒍𝒊𝒎 𝒙→𝟖 𝒏 𝒈(𝒙) d. 𝒍𝒊𝒎 𝒙→−𝟖 𝒈 𝒙 𝒏
Evaluate 𝒇 𝒙 = 𝟗𝒛 𝟐 − 𝟏𝟎𝟎, 𝒈 𝒙 = 𝒛 𝟐 − 𝟐𝟓 a. 𝒍𝒊𝒎 𝒛→−𝟑 𝒇 𝒙 𝒈 𝒙 b. 𝒍𝒊𝒎 𝒛→𝟑 𝒈 𝒙 𝒇 𝒙 Evaluate 𝒃 = 𝟑𝟔, 𝒇 𝒙 = 𝒙 𝟑 − 𝟐𝟕 a. 𝐥𝐢𝐦 𝒙→𝟑 𝒃 𝒇(𝒙) b. 𝐥𝐢𝐦 𝒙→𝟐 𝒃 𝒇(𝒙)
ANSWERS
1. Evaluate 𝒇 𝒙 = 𝟒 𝒙 𝟐 − 𝟔𝒙 + 𝟗, 𝒈 𝒙 = 𝒙 𝟑 + 𝟖𝒙 𝟐 − 𝟏𝟔𝒙 + 𝟒, 𝒏 = 𝟑; a. 𝒍𝒊𝒎 𝒙→𝟑 𝒏 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→𝟑 𝒏 )𝒇(𝒙 = 𝒏 𝒍𝒊𝒎 𝒙→𝟑 )𝒇(𝒙 = 𝟑 𝒍𝒊𝒎 𝒙→𝟑 𝟒 𝒙 𝟐 − 𝟔𝒙 + 𝟗 = 3 𝟒 𝟑 𝟐 − 𝟔 𝟑 + 𝟗 = 3 𝟑𝟔 − 𝟏𝟖 + 𝟗 = 𝟑= 3 𝟐𝟕
1. Evaluate 𝒇 𝒙 = 𝟒 𝒙 𝟐 − 𝟔𝒙 + 𝟗, 𝒈 𝒙 = 𝒙 𝟑 + 𝟖𝒙 𝟐 − 𝟏𝟔𝒙 + 𝟒, 𝒏 = 𝟑; b. 𝒍𝒊𝒎 𝒙→−𝟑 𝒇 𝒙 𝒏 𝒍𝒊𝒎 𝒙→−𝟑 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→−𝟑 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→−𝟑 𝟒𝒙 𝟐 − 𝟔𝒙 + 𝟗 𝟑 = 𝟒 −𝟑 𝟐 − 𝟔 −𝟑 + 𝟗 𝟑 = 𝟑𝟔 + 𝟏𝟖 + 𝟗 𝟑 = 𝟔𝟑 𝟑 = 𝟐𝟓𝟎, 𝟎𝟒𝟕
1. Evaluate 𝒇 𝒙 = 𝟒 𝒙 𝟐 − 𝟔𝒙 + 𝟗, 𝒈 𝒙 = 𝒙 𝟑 + 𝟖𝒙 𝟐 − 𝟏𝟔𝒙 + 𝟒, 𝒏 = 𝟑; c. 𝒍𝒊𝒎 𝒙→𝟐 𝒏 𝒈(𝒙) 𝒍𝒊𝒎 𝒙→𝟐 𝒏 )𝒈(𝒙 = 𝒏 𝒍𝒊𝒎 𝒙→𝟐 )𝒈(𝒙 = 𝟑 𝒍𝒊𝒎 𝒙→𝟐 𝒙 𝟑 + 𝟖𝒙 𝟐 − 𝟏𝟔𝒙 + 𝟒 = 3 𝟐 𝟑 + 𝟖 𝟐 𝟐 − 𝟏𝟔 𝟐 + 𝟒 = 3 𝟖 + 𝟑𝟐 − 𝟑𝟐 + 𝟒 = 𝟐. 𝟐𝟗= 3 𝟏𝟐
1. Evaluate 𝒇 𝒙 = 𝟒 𝒙 𝟐 − 𝟔𝒙 + 𝟗, 𝒈 𝒙 = 𝒙 𝟑 + 𝟖𝒙 𝟐 − 𝟏𝟔𝒙 + 𝟒, 𝒏 = 𝟑; d. 𝒍𝒊𝒎 𝒙→−𝟐 𝒈 𝒙 𝒏 𝒍𝒊𝒎 𝒙→−𝟐 𝒈 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→−𝟐 𝒈 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→−𝟐 𝒙 𝟑 + 𝟖𝒙 𝟐 − 𝟏𝟔𝒙 + 𝟒 𝟑 = −𝟐 𝟑 + 𝟖 −𝟐 𝟐 − 𝟏𝟔 −𝟐 + 𝟒 𝟑 = −𝟖 + 𝟑𝟐 + 𝟑𝟐 + 𝟒 𝟑 = 𝟔𝟎 𝟑 = 𝟐𝟏𝟔, 𝟎𝟎𝟎
2. Evaluate 𝒇 𝒙 = 𝟏𝟔𝐱 𝟐 − 𝟐𝟓, 𝒈 𝒙 = 𝒙 𝟑 − 𝟏 a. 𝒍𝒊𝒎 𝒙→𝟑 𝒇 𝒙 𝒈 𝒙 𝐥𝐢𝐦 𝒙→𝟑 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→3 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝟑 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝟑 𝟏𝟔𝒙 𝟐 − 𝟐𝟓 𝐥𝐢𝐦 𝒙→𝟑 𝐱 𝟑−𝟏 = 𝟏𝟔 𝟑 𝟐 − 𝟐𝟓 𝟑 𝟑−𝟏 = 𝟏𝟒𝟒 − 𝟐𝟓 𝟐𝟔 = 𝟏𝟏𝟗 𝟐𝟔 = 𝟗. 𝟐𝟏 × 𝟏𝟎 𝟓𝟑
2. Evaluate 𝒇 𝒙 = 𝟏𝟔𝐱 𝟐 − 𝟐𝟓, 𝒈 𝒙 = 𝒙 𝟑 − 𝟏 b. 𝒍𝒊𝒎 𝒙→𝟑 𝒈 𝒙 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝟑 𝒈 𝒙 𝒇 𝒙 = 𝐥𝐢𝐦 𝒙→3 𝒈 𝒙 𝐥𝐢𝐦 𝒙→𝟑 𝒇 𝒙 = 𝐥𝐢𝐦 𝒙→𝟑 𝒙 𝟑 − 𝟏 𝐥𝐢𝐦 𝒙→𝟑 𝟏𝟔𝐱 𝟐−𝟐𝟓 = ( 𝟑 𝟑 − 𝟏) 𝟏𝟔(𝟑) 𝟐−𝟐𝟓 = (𝟐𝟔) 𝟏𝟒𝟒−𝟐𝟓 = 𝟐𝟔 𝟏𝟏𝟗 = 𝟐. 𝟒𝟏 × 𝟏𝟎 𝟏𝟔𝟖
3. Evaluate 𝑏 = 𝟐𝟒, 𝑓 𝑥 = 𝒙 𝟑 − 𝟖 a. 𝐥𝐢𝐦 𝒙→𝟑 𝒃 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→𝟑 𝒃 )𝒇(𝒙 = 𝒍𝒊𝒎 𝒙→𝟑 𝟐𝟒 𝒙 𝟑−𝟖 = 𝒍𝒊𝒎 𝒙→𝟑 𝟐𝟒 𝒍𝒊𝒎 𝒙→𝟑 𝒙 𝟑−𝟖 = 𝟐𝟒 𝒙 𝟑−𝟖 = 𝟐𝟒 𝟑 𝟑−𝟖 = 𝟐𝟒 𝟏𝟗 = 𝟏. 𝟔𝟕 × 𝟏𝟎 𝟐𝟔
3. Evaluate 𝑏 = 𝟐𝟒, 𝑓 𝑥 = 𝒙 𝟑 − 𝟖 b. 𝐥𝐢𝐦 𝒙→𝟐 𝒃 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→𝟐 𝒃 )𝒇(𝒙 = 𝒍𝒊𝒎 𝒙→𝟐 𝟐𝟒 𝒙 𝟑−𝟖 = 𝒍𝒊𝒎 𝒙→𝟐 𝟐𝟒 𝒍𝒊𝒎 𝒙→𝟐 𝒙 𝟑−𝟖 = 𝟐𝟒 𝒙 𝟑−𝟖 = 𝟐𝟒 𝟐 𝟑−𝟖 = 𝟐𝟒 𝟎 = 𝟏
Limit of Logarithmic Function Let a and b be real numbers, where a > 0, b > 0 and b ≠ 1. Then, 𝒍𝒊𝒎 𝒙→𝒂 ( 𝒍𝒐𝒈 𝒃 𝒙) = 𝒍𝒐𝒈 𝒃 𝒂
Properties There are three basic properties in limits, which are used as formulas in evaluating the limits of logarithmic functions. 1.Product Rule 2.Quotient Rule 3.Power Rule
1. Product Rule Example: Solution: lim 𝑥→𝑎 (log 𝑏 𝐴𝐵) = lim 𝑥→𝑎 log 𝑏 𝐴 + log 𝑏 𝐵 Given 𝑨 = 𝒙 𝟐 + 𝟐𝒙 + 𝟏, 𝑩 = 𝒙 − 𝟓, 𝒃 = 𝟓 𝒂𝒏𝒅 𝒙 = 𝟑, find 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) 𝒍𝒊𝒎 𝒙→𝒂 𝐥𝐨𝐠 𝒃 𝑨𝑩 = 𝐥𝐢𝐦 𝒙→𝟑 𝐥𝐨𝐠 𝟓(𝒙 𝟐 +𝟐𝒙 + 𝟏) + 𝐥𝐨𝐠 𝟓 𝒙 − 𝟓) = 𝐥𝐨𝐠 𝟓 𝟑 𝟐 + 𝟐 𝟑 + 𝟏 + 𝟑 − 𝟓 = 𝐥𝐨𝐠 𝟓 𝟗 + 𝟔 + 𝟏 − 𝟐 = 𝐥𝐨𝐠 𝟓 𝟏𝟒 = 𝟏. 𝟔𝟒
2. Quotient Rule Example: Solution: lim 𝑥→𝑎 log 𝑏( 𝐴 𝐵 ) = lim 𝑥→𝑎 log 𝑏 𝐴 − log 𝑏 𝐵 Given 𝑨 = 𝒙 𝟐 + 𝟐𝒙 + 𝟏, 𝑩 = 𝒙 − 𝟓, 𝒃 = 𝟓 𝒂𝒏𝒅 𝒙 = 𝟑, find 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨 𝑩 ) 𝒍𝒊𝒎 𝒙→𝒂 𝐥𝐨𝐠 𝒃 𝑨 𝑩 = 𝐥𝐢𝐦 𝒙→𝟑 𝐥𝐨𝐠 𝟓(𝒙 𝟐 +𝟐𝒙 + 𝟏) − 𝐥𝐨𝐠 𝟓 𝒙 − 𝟓) = 𝐥𝐨𝐠 𝟓 𝟑 𝟐 + 𝟐 𝟑 + 𝟏 − 𝟑 − 𝟓 = 𝐥𝐨𝐠 𝟓 𝟗 + 𝟔 + 𝟏 + 𝟐 = 𝐥𝐨𝐠 𝟓 𝟏𝟖 = 𝟏. 𝟕𝟗𝟔
3. Power Rule Example: Solution: 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏 = 𝒍𝒊𝒎 𝒙→𝒂 𝒏 𝒍𝒐𝒈 𝒃 𝑨 Given 𝑨 = 𝒙 − 𝟓, 𝒏 = 𝟑, 𝒃 = 𝟐 𝒂𝒏𝒅 𝒙 = 𝟖, find 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏 𝒍𝒊𝒎 𝒙→𝟖 𝒍𝒐𝒈 𝟐 𝒙 − 𝟓 𝟑 = 𝒍𝒊𝒎 𝒙→𝟖 𝟑 𝒍𝒐𝒈 𝟐 𝒙 − 𝟓 = 𝟑 𝒍𝒐𝒈 𝟐 𝟖 − 𝟓 = 𝟑 𝒍𝒐𝒈 𝟐 𝟑 = 𝟑 𝟏. 𝟓𝟖𝟓 = 𝟒. 𝟕𝟓𝟓
Let’s Practice!!! 1. Given 𝑨 = 𝒙 𝟐 + 𝟒𝒙 + 𝟒, 𝑩 = 𝒙 𝟐 − 𝟏𝟔, 𝒙 = 𝟐, 𝒃 = 𝟏𝟎, 𝒏 = 𝟒 ; Find the following: a. 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏
Solution: 𝒍𝒊𝒎 𝒙→𝒂 𝐥𝐨𝐠 𝒃 𝑨𝑩 = 𝐥𝐢𝐦 𝒙→𝟐 𝐥𝐨𝐠 𝟏𝟎(𝒙 𝟐 + 𝟒𝒙 + 𝟒) + 𝐥𝐨𝐠 𝟏𝟎(𝒙 𝟐 − 𝟏𝟔) = 𝐥𝐨𝐠 𝟏𝟎 𝟐 𝟐 + 𝟒 𝟐 + 𝟒 + 𝟐 𝟐 − 𝟏𝟔 = 𝐥𝐨𝐠 𝟏𝟎 𝟒 + 𝟖 + 𝟒 + 𝟒 − 𝟏𝟔 = 𝐥𝐨𝐠 𝟏𝟎 𝟒 = 𝟎. 𝟔𝟎𝟐 1. Given 𝑨 = 𝒙 𝟐 + 𝟒𝒙 + 𝟒, 𝑩 = 𝒙 𝟐 − 𝟏𝟔, 𝒙 = 𝟐, 𝒃 = 𝟏𝟎, 𝒏 = 𝟒 ;
Solution: 𝒍𝒊𝒎 𝒙→𝒂 𝐥𝐨𝐠 𝒃 𝑨 𝑩 = 𝐥𝐢𝐦 𝒙→𝟐 𝐥𝐨𝐠 𝟏𝟎 𝒙 𝟐 + 𝟒𝒙 + 𝟒 − 𝐥𝐨𝐠 𝟏𝟎(𝒙 𝟐 − 𝟏𝟔) 1. Given 𝑨 = 𝒙 𝟐 + 𝟒𝒙 + 𝟒, 𝑩 = 𝒙 𝟐 − 𝟏𝟔, 𝒙 = 𝟐, 𝒃 = 𝟏𝟎, 𝒏 = 𝟒 ; = 𝐥𝐨𝐠 𝟏𝟎 𝟐 𝟐 + 𝟒 𝟐 + 𝟒 − 𝟐 𝟐 − 𝟏𝟔 = 𝐥𝐨𝐠 𝟏𝟎 𝟒 + 𝟖 + 𝟒 − 𝟒 + 𝟏𝟔 = 𝐥𝐨𝐠 𝟏𝟎 𝟐𝟖 = 𝟏. 𝟒𝟒𝟕
Solution: 𝒍𝒊𝒎 𝒙→𝟐 𝒍𝒐𝒈 𝟏𝟎 𝒙 𝟐 + 𝟒𝒙 + 𝟒 𝟒 = 𝒍𝒊𝒎 𝒙→𝟐 𝟒 𝒍𝒐𝒈 𝟏𝟎 𝒙 𝟐 + 𝟒𝒙 + 𝟒 = 𝟒 𝒍𝒐𝒈 𝟏𝟎 𝟐 𝟐 + 𝟒 𝟐 + 𝟒 = 𝟒 𝒍𝒐𝒈 𝟏𝟎 𝟏𝟔 = 𝟒 𝟏. 𝟐𝟎𝟒 = 𝟒. 𝟖𝟏𝟔 1. Given 𝑨 = 𝒙 𝟐 + 𝟒𝒙 + 𝟒, 𝑩 = 𝒙 𝟐 − 𝟏𝟔, 𝒙 = 𝟐, 𝒃 = 𝟏𝟎, 𝒏 = 𝟒 ;
Seatwork 3.2
1. Given 𝑨 = 𝟗𝒙 𝟐 + 𝟑𝟎𝒙 + 𝟐𝟓, 𝑩 = 𝟒𝒙 𝟐 − 𝟏𝟔, 𝒙 = 𝟓, 𝒃 = 𝟑𝟓, 𝒏 = 𝟏𝟓 ; 2. Given 𝑨 = 𝒙 𝟐 − 𝟏𝟔𝒙 + 𝟔𝟒, 𝑩 = 𝒙 𝟐 − 𝟏, 𝒙 = 𝟑, 𝒃 = 𝟐𝟓, 𝒏 = 𝟐𝟓 Find the following: a. 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏
2. Given 𝑨 = 𝒙 𝟐 + 𝟐𝟎𝒙 − 𝟏𝟎𝟎, 𝑩 = 𝒙 𝟐 − 𝟑𝟔, 𝒙 = 𝟏𝟐, 𝒃 = 𝟑𝟔, 𝒏 = 𝟐𝟓 ; Find the following: a. 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏
ANSWERS
Solution: 𝒍𝒊𝒎 𝒙→𝒂 𝐥𝐨𝐠 𝒃 𝑨𝑩 = 𝐥𝐢𝐦 𝒙→𝟐 𝐥𝐨𝐠 𝟏𝟓(𝟐𝟓𝒙 𝟐 + 𝟒𝟎𝒙 + 𝟏𝟔) + 𝐥𝐨𝐠 𝟏𝟓(𝒙 𝟐 − 𝟔𝟒) = 𝐥𝐨𝐠 𝟏𝟓 𝟐𝟓 𝟐 𝟐 + 𝟒𝟎 𝟐 + 𝟏𝟔 + 𝟐 𝟐 − 𝟔𝟒 = 𝐥𝐨𝐠 𝟏𝟓 𝟏𝟎𝟎 + 𝟖𝟎 + 𝟏𝟔 + 𝟒 − 𝟔𝟒 = 𝐥𝐨𝐠 𝟏𝟓 𝟏𝟑𝟔 = 𝟏. 𝟖𝟏𝟒 1. Given 𝑨 = 𝟐𝟓𝒙 𝟐 + 𝟒𝟎𝒙 + 𝟏𝟔, 𝑩 = 𝒙 𝟐 − 𝟔𝟒, 𝒙 = 𝟐, 𝒃 = 𝟏𝟓, 𝒏 = 𝟓 ;
Solution: 𝒍𝒊𝒎 𝒙→𝒂 𝐥𝐨𝐠 𝒃 𝑨 𝑩 = 𝐥𝐢𝐦 𝒙→𝟐 𝐥𝐨𝐠 𝟏𝟓 𝟐𝟓𝒙 𝟐 + 𝟒𝟎𝒙 + 𝟏𝟔 − 𝐥𝐨𝐠 𝟏𝟓(𝒙 𝟐 − 𝟔𝟒) 1. Given 𝑨 = 𝟐𝟓𝒙 𝟐 + 𝟒𝟎𝒙 + 𝟏𝟔, 𝑩 = 𝒙 𝟐 − 𝟔𝟒, 𝒙 = 𝟐, 𝒃 = 𝟏𝟓, 𝒏 = 𝟓 ; = 𝐥𝐨𝐠 𝟏𝟓 𝟐𝟓 𝟐 𝟐 + 𝟒𝟎 𝟐 + 𝟏𝟔 − 𝟐 𝟐 − 𝟔𝟒 = 𝐥𝐨𝐠 𝟏𝟓 𝟏𝟎𝟎 + 𝟖𝟎 + 𝟏𝟔 − 𝟒 + 𝟔𝟒 = 𝐥𝐨𝐠 𝟏𝟓 𝟐𝟓𝟔 = 𝟐. 𝟎𝟒𝟖
Solution: 𝒍𝒊𝒎 𝒙→𝟐 𝒍𝒐𝒈 𝟏𝟓 𝟐𝟓𝒙 𝟐 + 𝟒𝟎𝒙 + 𝟏𝟔 𝟓 = 𝒍𝒊𝒎 𝒙→𝟐 𝟓 𝒍𝒐𝒈 𝟏𝟓 𝟐𝟓𝒙 𝟐 + 𝟒𝟎𝒙 + 𝟏𝟔 = 𝟓 𝒍𝒐𝒈 𝟏𝟓 𝟐𝟓 𝟐 𝟐 + 𝟒𝟎 𝟐 + 𝟏𝟔 = 𝟓 𝒍𝒐𝒈 𝟏𝟓 𝟏𝟗𝟔 = 𝟓 𝟏. 𝟗𝟒𝟗 = 𝟗. 𝟕𝟒𝟓 1. Given 𝑨 = 𝟐𝟓𝒙 𝟐 + 𝟒𝟎𝒙 + 𝟏𝟔, 𝑩 = 𝒙 𝟐 − 𝟔𝟒, 𝒙 = 𝟐, 𝒃 = 𝟏𝟓, 𝒏 = 𝟓 ;
Solution: 𝒍𝒊𝒎 𝒙→𝒂 𝐥𝐨𝐠 𝒃 𝑨𝑩 = 𝐥𝐢𝐦 𝒙→𝟏𝟐 𝐥𝐨𝐠 𝟑𝟔(𝒙 𝟐 + 𝟐𝟎𝒙 − 𝟏𝟎𝟎) + 𝐥𝐨𝐠 𝟑𝟔(𝒙 𝟐 − 𝟑𝟔) = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟐 𝟐 + 𝟐𝟎 𝟏𝟐 − 𝟏𝟎𝟎 + 𝟏𝟐 𝟐 − 𝟑𝟔 = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟒𝟒 + 𝟐𝟒𝟎 − 𝟏𝟎𝟎 + 𝟏𝟒𝟒 − 𝟑𝟔 = 𝐥𝐨𝐠 𝟑𝟔 𝟑𝟗𝟐 = 𝟏. 𝟔𝟔𝟔 2. Given 𝑨 = 𝒙 𝟐 + 𝟐𝟎𝒙 − 𝟏𝟎𝟎, 𝑩 = 𝒙 𝟐 − 𝟑𝟔, 𝒙 = 𝟏𝟐, 𝒃 = 𝟑𝟔, 𝒏 = 𝟐𝟓 ;
Solution: 𝒍𝒊𝒎 𝒙→𝒂 𝐥𝐨𝐠 𝒃 𝑨 𝑩 = 𝐥𝐢𝐦 𝒙→𝟏𝟐 𝐥𝐨𝐠 𝟑𝟔 𝒙 𝟐 + 𝟐𝟎𝒙 − 𝟏𝟎𝟎 − 𝐥𝐨𝐠 𝟑𝟔(𝒙 𝟐 − 𝟑𝟔) 2. Given 𝑨 = 𝒙 𝟐 + 𝟐𝟎𝒙 − 𝟏𝟎𝟎, 𝑩 = 𝒙 𝟐 − 𝟑𝟔, 𝒙 = 𝟏𝟐, 𝒃 = 𝟑𝟔, 𝒏 = 𝟐𝟓 ; = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟐 𝟐 + 𝟐𝟎 𝟏𝟐 − 𝟏𝟎𝟎 − 𝟏𝟐 𝟐 − 𝟑𝟔 = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟒𝟒 + 𝟐𝟒𝟎 − 𝟏𝟎𝟎 − 𝟏𝟒𝟒 + 𝟑𝟔 = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟕𝟔 = 𝟏. 𝟒𝟒𝟑
Solution: 𝒍𝒊𝒎 𝒙→𝟏𝟐 𝒍𝒐𝒈 𝟑𝟔 𝒙 𝟐 + 𝟐𝟎𝒙 − 𝟏𝟎𝟎 𝟐𝟓 = 𝒍𝒊𝒎 𝒙→𝟏𝟐 𝟐𝟓 𝒍𝒐𝒈 𝟑𝟔 𝒙 𝟐 + 𝟐𝟎𝒙 − 𝟏𝟎𝟎 = 𝟐𝟓 𝒍𝒐𝒈 𝟑𝟔 𝟏𝟐 𝟐 + 𝟐𝟎 𝟏𝟐 − 𝟏𝟎𝟎 = 𝟐𝟓 𝒍𝒐𝒈 𝟑𝟔 𝟐𝟖𝟒 = 𝟐𝟓 𝟏. 𝟓𝟕𝟔 = 𝟑𝟗. 𝟒𝟎𝟗 2. Given 𝑨 = 𝒙 𝟐 + 𝟐𝟎𝒙 − 𝟏𝟎𝟎, 𝑩 = 𝒙 𝟐 − 𝟑𝟔, 𝒙 = 𝟏𝟐, 𝒃 = 𝟑𝟔, 𝒏 = 𝟐𝟓 ;
Quiz 3.2
Given 𝑨 = 𝟑𝟔𝒚 𝟐 − 𝟗𝟔𝒚 + 𝟔𝟒, 𝑩 = 𝒚 𝟐 − 𝟔𝟒, 𝒚 = 𝟓, 𝒃 = 𝟐𝟓, 𝒏 = 𝟖 ; Find the following: a. 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏
Given 𝑨 = 𝟒𝟓𝒎 𝟐 − 𝟑𝟔𝒎 − 𝟗, 𝑩 = 𝒎 𝟐 − 𝟐𝒎 + 𝟐, 𝒎 = 𝟔, 𝒃 = 𝟑𝟔, 𝒏 = 𝟓 ; Find the following: a. 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏
Given 𝑨 = 𝟏𝟔𝒚 𝟐 − 𝟒𝟖𝒚 + 𝟑𝟔, 𝑩 = 𝒚 𝟑 − 𝟏, 𝒚 = 𝟑, 𝒃 = 𝟏𝟖, 𝒏 = 𝟔 ; Find the following: a. 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏
ANSWERS
Solution: 𝒍𝒊𝒎 𝐲→𝒂 𝐥𝐨𝐠 𝒃 𝑨𝑩 = 𝐥𝐢𝐦 𝐲→𝟓 𝐥𝐨𝐠 𝟐𝟓(𝟑𝟔𝒚 𝟐 − 𝟗𝟔𝒚 + 𝟔𝟒) + 𝐥𝐨𝐠 𝟐𝟓(𝒚 𝟐 − 𝟔𝟒) = 𝐥𝐨𝐠 𝟐𝟓 𝟑𝟔 𝟓 𝟐 − 𝟗𝟔 𝟓 + 𝟔𝟒 + 𝟓 𝟐 − 𝟔𝟒 = 𝐥𝐨𝐠 𝟐𝟓 𝟗𝟎𝟎 − 𝟒𝟖𝟎 + 𝟔𝟒 + 𝟐𝟓 − 𝟔𝟒 = 𝐥𝐨𝐠 𝟐𝟓 𝟒𝟒𝟓 = 𝟏. 𝟖𝟗𝟒 1. Given 𝑨 = 𝟑𝟔𝒚 𝟐 − 𝟗𝟔𝒚 + 𝟔𝟒, 𝑩 = 𝒚 𝟐 − 𝟔𝟒, 𝒚 = 𝟓, 𝒃 = 𝟐𝟓, 𝒏 = 𝟖
Solution: = 𝟏. 𝟗𝟒𝟓 1. Given 𝑨 = 𝟑𝟔𝒚 𝟐 − 𝟗𝟔𝒚 + 𝟔𝟒, 𝑩 = 𝒚 𝟐 − 𝟔𝟒, 𝒚 = 𝟓, 𝒃 = 𝟐𝟓, 𝒏 = 𝟖 𝒍𝒊𝒎 𝐲→𝒂 𝐥𝐨𝐠 𝒃 𝑨 𝑩 = 𝐥𝐢𝐦 𝐲→𝟓 𝐥𝐨𝐠 𝟐𝟓 𝟑𝟔𝒚 𝟐 − 𝟗𝟔𝒚 + 𝟔𝟒 − 𝐥𝐨𝐠 𝟐𝟓(𝒚 𝟐 − 𝟔𝟒) = 𝐥𝐨𝐠 𝟐𝟓 𝟑𝟔 𝟓 𝟐 − 𝟗𝟔 𝟓 + 𝟔𝟒 − 𝟓 𝟐 − 𝟔𝟒 = 𝐥𝐨𝐠 𝟐𝟓 𝟗𝟎𝟎 − 𝟒𝟖𝟎 + 𝟔𝟒 − 𝟐𝟓 + 𝟔𝟒 = 𝐥𝐨𝐠 𝟐𝟓 𝟓𝟐𝟑
Solution: 𝒍𝒊𝒎 𝒚→𝟓 𝒍𝒐𝒈 𝟐𝟓 𝟑𝟔𝒚 𝟐 − 𝟗𝟔𝒚 + 𝟔𝟒 𝟖 = 𝒍𝒊𝒎 𝒚→𝟓 𝟖 𝒍𝒐𝒈 𝟐𝟓 𝟑𝟔𝒚 𝟐 − 𝟗𝟔𝒚 + 𝟔𝟒 = 𝟖 𝒍𝒐𝒈 𝟐𝟓 𝟑𝟔 𝟓 𝟐 − 𝟗𝟔 𝟓 + +𝟔𝟒 = 𝟖 𝒍𝒐𝒈 𝟐𝟓 𝟒𝟖𝟒 = 𝟖 𝟏. 𝟗𝟐𝟏 = 𝟏𝟓. 𝟑𝟔𝟓 1. Given 𝑨 = 𝟑𝟔𝒚 𝟐 − 𝟗𝟔𝒚 + 𝟔𝟒, 𝑩 = 𝒚 𝟐 − 𝟔𝟒, 𝒚 = 𝟓, 𝒃 = 𝟐𝟓, 𝒏 = 𝟖
Solution: 𝒍𝒊𝒎 𝐦→𝒂 𝐥𝐨𝐠 𝒃 𝑨𝑩 = 𝐥𝐢𝐦 𝐦→𝟔 𝐥𝐨𝐠 𝟑𝟔(𝟒𝟓𝒎 𝟐 − 𝟑𝟔𝒎 − 𝟗) + 𝐥𝐨𝐠 𝟑𝟔(𝒎 𝟐 − 𝟐𝒎 + 𝟐) = 𝐥𝐨𝐠 𝟑𝟔 𝟒𝟓 𝟔 𝟐 − 𝟑𝟔 𝟔 − 𝟗 + 𝟔 𝟐 − 𝟐 𝟔 + 𝟐 = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟔𝟐𝟎 − 𝟐𝟏𝟔 − 𝟗 + 𝟑𝟔 − 𝟏𝟐 + 𝟐 = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟒𝟐𝟏 = 𝟐. 𝟎𝟐𝟔 2. Given 𝑨 = 𝟒𝟓𝒎 𝟐 − 𝟑𝟔𝒎 − 𝟗, 𝑩 = 𝒎 𝟐 − 𝟐𝒎 + 𝟐, 𝒎 = 𝟔, 𝒃 = 𝟑𝟔, 𝒏 = 𝟓 ;
Solution: = 𝟐. 𝟎𝟏𝟓 2. Given 𝑨 = 𝟒𝟓𝒎 𝟐 − 𝟑𝟔𝒎 − 𝟗, 𝑩 = 𝒎 𝟐 − 𝟐𝒎 + 𝟐, 𝒎 = 𝟔, 𝒃 = 𝟑𝟔, 𝒏 = 𝟓 ; 𝒍𝒊𝒎 𝐦→𝒂 𝐥𝐨𝐠 𝒃 𝑨 𝑩 = 𝐥𝐢𝐦 𝐦→𝟔 𝐥𝐨𝐠 𝟑𝟔 𝟒𝟓𝒎 𝟐 − 𝟑𝟔𝒎 − 𝟗 − 𝐥𝐨𝐠 𝟑𝟔(𝒎 𝟐 − 𝟐𝒎 + 𝟐) = 𝐥𝐨𝐠 𝟑𝟔 𝟒𝟓 𝟔 𝟐 − 𝟑𝟔 𝟔 − 𝟗 − 𝟔 𝟐 − 𝟐 𝟔 + 𝟐 = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟔𝟐𝟎 − 𝟐𝟏𝟔 − 𝟗 − 𝟑𝟔 + 𝟏𝟐 − 𝟐 = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟑𝟔𝟗
Solution: 𝒍𝒊𝒎 𝒎→𝟔 𝒍𝒐𝒈 𝟑𝟔 𝟒𝟓𝒎 𝟐 − 𝟑𝟔𝒎 − 𝟗 𝟓 = 𝒍𝒊𝒎 𝒙→𝟐 𝟓 𝒍𝒐𝒈 𝟑𝟔 𝟒𝟓𝒎 𝟐 − 𝟑𝟔𝒎 − 𝟗 = 𝟓 𝒍𝒐𝒈 𝟑𝟔 𝟒𝟓 𝟔 𝟐 − 𝟑𝟔 𝟔 − 𝟗 = 𝟓 𝒍𝒐𝒈 𝟑𝟔 𝟏𝟑𝟗𝟓 = 𝟓 𝟐. 𝟎𝟐𝟏 = 𝟏𝟎. 𝟏𝟎𝟑 2. Given 𝑨 = 𝟒𝟓𝒎 𝟐 − 𝟑𝟔𝒎 − 𝟗, 𝑩 = 𝒎 𝟐 − 𝟐𝒎 + 𝟐, 𝒎 = 𝟔, 𝒃 = 𝟑𝟔, 𝒏 = 𝟓 ;
Solution: 𝒍𝒊𝒎 𝐲→𝒂 𝐥𝐨𝐠 𝒃 𝑨𝑩 = 𝐥𝐢𝐦 𝒚→𝟑 𝐥𝐨𝐠 𝟏𝟖(𝟏𝟔𝒚 𝟐 − 𝟒𝟔𝒚 + 𝟑𝟔) + 𝐥𝐨𝐠 𝟏𝟖(𝒚 𝟑 − 𝟏) = 𝐥𝐨𝐠 𝟏𝟖 𝟏𝟔 𝟑 𝟐 − 𝟒𝟖 𝟑 + 𝟑𝟔 + 𝟑 𝟑 − 𝟏 = 𝐥𝐨𝐠 𝟏𝟖 𝟏𝟒𝟒 − 𝟏𝟒𝟒 + 𝟑𝟔 + 𝟐𝟕 − 𝟏 = 𝐥𝐨𝐠 𝟏𝟖 𝟔𝟐 = 𝟏. 𝟒𝟐𝟖 3. Given 𝑨 = 𝟏𝟔𝒚 𝟐 − 𝟒𝟖𝒚 + 𝟑𝟔, 𝑩 = 𝒚 𝟑 − 𝟏, 𝒚 = 𝟑, 𝒃 = 𝟏𝟖, 𝒏 = 𝟔 ;
Solution: = 𝟎. 𝟕𝟗𝟕 3. Given 𝑨 = 𝟏𝟔𝒚 𝟐 − 𝟒𝟖𝒚 + 𝟑𝟔, 𝑩 = 𝒚 𝟑 − 𝟏, 𝒚 = 𝟑, 𝒃 = 𝟏𝟖, 𝒏 = 𝟔 ; 𝒍𝒊𝒎 𝐲→𝒂 𝐥𝐨𝐠 𝒃 𝑨 𝑩 = 𝐥𝐢𝐦 𝒚→𝟑 𝐥𝐨𝐠 𝟏𝟖 𝟏𝟔𝒚 𝟐 − 𝟒𝟔𝒚 + 𝟑𝟔 − 𝐥𝐨𝐠 𝟏𝟖(𝒚 𝟑 − 𝟏) = 𝐥𝐨𝐠 𝟏𝟖 𝟏𝟔 𝟑 𝟐 − 𝟒𝟖 𝟑 + 𝟑𝟔 − 𝟑 𝟐 − 𝟏 = 𝐥𝐨𝐠 𝟏𝟖 𝟏𝟒𝟒 − 𝟏𝟒𝟒 + 𝟑𝟔 − 𝟐𝟕 + 𝟏 = 𝐥𝐨𝐠 𝟏𝟖 𝟏𝟎
Solution: 𝒍𝒊𝒎 𝒚→𝟑 𝒍𝒐𝒈 𝟏𝟖 𝟏𝟔𝒚 𝟐 − 𝟒𝟖𝒚 + 𝟑𝟔 𝟔 = 𝒍𝒊𝒎 𝒚→𝟑 𝟔 𝒍𝒐𝒈 𝟏𝟖 𝟏𝟔𝒚 𝟐 − 𝟒𝟖𝒚 + 𝟑𝟔 = 𝟔 𝒍𝒐𝒈 𝟏𝟖 𝟏𝟔 𝟑 𝟐 − 𝟒𝟖 𝟑 + 𝟑𝟔 = 𝟔 𝒍𝒐𝒈 𝟏𝟖 𝟑𝟔 = 𝟔 𝟏. 𝟐𝟒𝟎 = 𝟕. 𝟒𝟑𝟗 3. Given 𝑨 = 𝟏𝟔𝒚 𝟐 − 𝟒𝟖𝒚 + 𝟑𝟔, 𝑩 = 𝒚 𝟑 − 𝟏, 𝒚 = 𝟑, 𝒃 = 𝟏𝟖, 𝒏 = 𝟔 ;
Pointers to Review Limits of Non – Algebraic Functions Limit of an Exponential Function Limit of Logarithmic Function
Given 𝑨 = 𝒚 𝟐 + 𝟔𝒚 + 𝟗, 𝑩 = 𝟒𝒚 𝟐 − 𝟏, 𝒚 = 𝟖, 𝒃 = 𝟏𝟓, 𝒏 = 𝟔 ; Find the following: a. 𝒍𝒊𝒎 𝒚→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒚→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒚→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏
Given 𝑨 = 𝟑𝒙 𝟐 + 𝟗𝒙 + 𝟗, 𝑩 = 𝒙 − 𝟓, 𝒙 = 𝟐, 𝒃 = 𝟏𝟑, 𝒏 = 𝟏𝟎 ; Find the following: a. 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏
Given 𝑨 = 𝟔𝒛 𝟐 − 𝟒𝒛 + 𝟓, 𝑩 = 𝟐𝒛 𝟐 + 𝟒𝒛 − 𝟑, 𝒛 = 𝟑, 𝒃 = 𝟖, 𝒏 = 𝟐; Find the following: a. 𝒍𝒊𝒎 𝒛→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒛→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒛→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏

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世界を変えるクレーンを生み出そう！　高知エンジニアリングキャンプ2024プログラム
世界を変えるクレーンを生み出そう！　高知エンジニアリングキャンプ2024プログラム世界を変えるクレーンを生み出そう！　高知エンジニアリングキャンプ2024プログラム
世界を変えるクレーンを生み出そう！　高知エンジニアリングキャンプ2024プログラム
 

Basic Calculus Lesson 3

  • 1.
  • 2. Limit of an Exponential Function Let a and b be real numbers, where b > 0 and b ≠ 1. Then, 𝐥𝐢𝐦 𝒙→𝒂 𝒃 𝒙 = 𝒃 𝒂 .
  • 3. Properties There are four basic properties in limits, which are used as formulas in evaluating the limits of exponential functions. 1.Power Rule 2.Constant Base Power Rule 3.Constant Exponent Power Rule 4.Radical Power Rule
  • 4. 1. Power Rule 𝒍𝒊𝒎 𝒙→𝒂 𝒇 𝒙 𝒈 𝒙 = 𝒍𝒊𝒎 𝒙→𝒂 𝒇 𝒙 𝒍𝒊𝒎 𝒙→𝒂 𝒈 𝒙 It is a property of power rule, used to find the limit of an exponential function whose base and exponent are in a function form.
  • 5. 1. Power Rule 𝒍𝒊𝒎 𝒙→𝒂 𝒇 𝒙 𝒈 𝒙 = 𝒍𝒊𝒎 𝒙→𝒂 𝒇 𝒙 𝒍𝒊𝒎 𝒙→𝒂 𝒈 𝒙 Example: Evaluate 𝒇 𝒙 = 𝒙 𝟐 − 𝟒, 𝒈 𝒙 = 𝟐𝒙 ; 𝐥𝐢𝐦 𝒙→𝟑 𝒇 𝒙 𝒈 𝒙 Solution: 𝐥𝐢𝐦 𝒙→3 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→3 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝟑 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→3 𝒇 𝒙2 − 4 𝐥𝐢𝐦 𝒙→3 𝒈 2𝒙 = 𝟑 𝟐 − 𝟒 𝟐 𝟑 = 𝟗 − 𝟒 𝟔 = 𝟓 𝟔 = 𝟏𝟓, 𝟔𝟐𝟓
  • 6. 2. Constant Base Power Rule 𝒍𝒊𝒎 𝒙→𝒂 𝒃 𝒇 𝒙 = 𝒍𝒊𝒎 𝒙→𝒂 𝒃 𝒍𝒊𝒎 𝒙→𝒂 𝒇 𝒙 The limit of an exponential function is equal to the limit of the exponent with same base. It is called the limit rule of an exponential function.
  • 7. 2. Constant Base Power Rule 𝒍𝒊𝒎 𝒙→𝒂 𝒃 𝒇 𝒙 = 𝒍𝒊𝒎 𝒙→𝒂 𝒃 𝒍𝒊𝒎 𝒙→𝒂 𝒇 𝒙 Example: Solution: Evaluate 𝑏 = 12, 𝑓 𝑥 = 𝑥 + 3; 𝐥𝐢𝐦 𝒙→𝟏 𝒃 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→𝟏 𝒃 )𝒇(𝒙 = 𝒍𝒊𝒎 𝒙→𝟏 𝟏𝟐 𝒙+𝟑 = 𝒍𝒊𝒎 𝒙→𝟏 𝟏𝟐 𝒍𝒊𝒎 𝒙→𝟏 𝒙+𝟑 = 𝟏𝟐 𝒙+𝟑 = 𝟏𝟐 𝟏+𝟑 = 𝟏𝟐 𝟒 = 𝟐𝟎, 𝟕𝟑𝟔
  • 8. 3. Constant Exponent Power Rule 𝒍𝒊𝒎 𝒙→𝒂 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→𝒂 𝒇 𝒙 𝒏 Example: Solution: Evaluate 𝑓 𝑥 = 16𝑥2 − 64, 𝑛 = 2; 𝐥𝐢𝐦 𝒙→𝟏 𝒇 𝒙 𝒏 𝒍𝒊𝒎 𝒙→𝟏 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→𝟏 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→𝟏 𝟏𝟔𝒙 𝟐 − 𝟔𝟒 𝟐 = 𝟏𝟔 𝟏 𝟐 − 𝟔𝟒 𝟐 = 𝟏𝟔 − 𝟔𝟒 𝟐 = −𝟒𝟖 𝟐 = 𝟐, 𝟑𝟎𝟒
  • 9. 4. Radical Power Rule 𝒍𝒊𝒎 𝒙→𝒂 𝒏 )𝒇(𝒙 = 𝒏 𝒍𝒊𝒎 𝒙→𝒂 )𝒇(𝒙 Example: Solution: Evaluate 𝒇 𝒙 = 𝒙 𝟐 − 𝟔𝒙 + 𝟗, 𝒏 = 𝟐; 𝒍𝒊𝒎 𝒙→𝟔 𝒏 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→𝟔 𝒏 )𝒇(𝒙 = 𝒏 𝒍𝒊𝒎 𝒙→𝟔 )𝒇(𝒙 = 𝐥𝐢𝐦 𝒙→6 𝒙2 − 6𝒙 + 9 = 𝒙 − 3 2 = 𝒙 − 3 = 6 − 3 = 𝟑
  • 10. Let’s Practice!!! 1. Evaluate 𝒇 𝒙 = 𝟒 𝒙 𝟐 − 𝟐𝟒𝒙 + 𝟑𝟔, 𝒏 = 𝟐; a. 𝒍𝒊𝒎 𝒙→−𝟑 𝒏 𝒇(𝒙) b. 𝒍𝒊𝒎 𝒙→𝟒 𝒇 𝒙 𝒏 2. Evaluate 𝒇 𝒙 = 𝒙 𝟑 − 𝟐𝟕, 𝒈 𝒙 = 𝟐𝒙 + 𝟏 a. 𝒍𝒊𝒎 𝒙→𝟐 𝒇 𝒙 𝒈 𝒙 3. Evaluate 𝑏 = 𝟑, 𝑓 𝑥 = 𝒙 − 𝟒 a. 𝐥𝐢𝐦 𝒙→𝟖 𝒃 𝒇(𝒙)
  • 11. 1. Evaluate 𝒇 𝒙 = 𝟒 𝒙 𝟐 − 𝟐𝟒𝒙 + 𝟑𝟔, 𝒏 = 𝟐; a. 𝒍𝒊𝒎 𝒙→−𝟑 𝒏 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→−𝟑 𝒏 )𝒇(𝒙 = 𝒏 𝒍𝒊𝒎 𝒙→−𝟑 )𝒇(𝒙 = 𝐥𝐢𝐦 𝒙→−𝟑 𝟒𝒙2 − 𝟐𝟒𝒙 + 𝟑𝟔 = 𝟐𝒙 − 𝟔 2 = 𝟐𝒙 − 𝟔 = 𝟐 −𝟑 − 𝟔 = −𝟏𝟐 = −𝟔 − 𝟔
  • 12. 1. Evaluate 𝒇 𝒙 = 𝟒 𝒙 𝟐 − 𝟐𝟒𝒙 + 𝟑𝟔, 𝒏 = 𝟐; b. 𝒍𝒊𝒎 𝒙→𝟒 𝒇 𝒙 𝒏 𝒍𝒊𝒎 𝒙→𝟒 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→𝟒 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→𝟒 𝟒𝒙 𝟐 − 𝟐𝟒𝒙 + 𝟑𝟔 𝟐 = 𝟒 𝟒 𝟐 − 𝟐𝟒 𝟒 + 𝟑𝟔 𝟐 = 𝟒 𝟏𝟔 − 𝟗𝟔 + 𝟑𝟔 𝟐 = 𝟒 𝟐 = 𝟏𝟔 = 𝟔𝟒 − 𝟗𝟔 + 𝟑𝟔 𝟐
  • 13. 2. Evaluate 𝒇 𝒙 = 𝒙 𝟑 − 𝟐𝟕, 𝒈 𝒙 = 𝟐𝒙 + 𝟏 a. 𝒍𝒊𝒎 𝒙→𝟐 𝒇 𝒙 𝒈 𝒙 𝐥𝐢𝐦 𝒙→𝟐 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→2 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝟐 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝟐 𝒙 𝟑 − 𝟐𝟕 𝐥𝐢𝐦 𝒙→𝟐 𝟐𝐱+𝟏 = 𝟐 𝟑 − 𝟐𝟕 𝟐 𝟐 +𝟏 = 𝟖 − 𝟐𝟕 𝟓 = −𝟏𝟗 𝟓 = −𝟐, 𝟒𝟕𝟔, 𝟎𝟗𝟗
  • 14. 3. Evaluate 𝑏 = 𝟑, 𝑓 𝑥 = 𝒙 − 𝟒 a. 𝐥𝐢𝐦 𝒙→𝟖 𝒃 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→𝟖 𝒃 )𝒇(𝒙 = 𝒍𝒊𝒎 𝒙→𝟖 𝟑 𝒙−𝟒 = 𝒍𝒊𝒎 𝒙→𝟖 𝟑 𝒍𝒊𝒎 𝒙→𝟖 𝒙−𝟒 = 𝟑 𝒙−𝟒 = 𝟑 𝟖−𝟒 = 𝟑 𝟒 = 𝟖𝟏
  • 16. 1. Evaluate 𝒇 𝒙 = 𝒙 𝟐 + 𝟏𝟔𝒙 + 𝟖, 𝒏 = 𝟑; a. 𝒍𝒊𝒎 𝒙→𝟒 𝒏 𝒇(𝒙) b. 𝒍𝒊𝒎 𝒙→−𝟐 𝒇 𝒙 𝒏 2. Evaluate 𝒇 𝒙 = 𝒙 𝟐 − 𝟔𝟒, 𝒈 𝒙 = 𝒙 − 𝟓 a. 𝒍𝒊𝒎 𝒙→𝟗 𝒇 𝒙 𝒈 𝒙 b. 𝒍𝒊𝒎 𝒙→𝟗 𝒈 𝒙 𝒇 𝒙 3. Evaluate 𝑏 = 𝟏𝟓, 𝑓 𝑥 = 𝒙 𝟐 + 𝟐𝒙 + 𝟏 a. 𝐥𝐢𝐦 𝒙→𝟐 𝒃 𝒇(𝒙) b. 𝐥𝐢𝐦 𝒙→−𝟑 𝒃 𝒇(𝒙)
  • 18. 1. Evaluate 𝒇 𝒙 = 𝒙 𝟐 + 𝟏𝟔𝒙 + 𝟖, 𝒏 = 𝟑; a. 𝒍𝒊𝒎 𝒙→𝟒 𝒏 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→𝟒 𝒏 )𝒇(𝒙 = 𝒏 𝒍𝒊𝒎 𝒙→𝟒 )𝒇(𝒙 = 𝟑 𝒍𝒊𝒎 𝒙→𝟒 𝒙 𝟐 + 𝟏𝟔𝒙 + 𝟖 = 3 𝟒 𝟐 + 𝟏𝟔 𝟒 + 𝟖 = 3 𝟏𝟔 + 𝟔𝟒 + 𝟖 = 𝟒. 𝟒𝟓 = 3 𝟖𝟖
  • 19. 1. Evaluate 𝒇 𝒙 = 𝒙 𝟐 + 𝟏𝟔𝒙 + 𝟖, 𝒏 = 𝟑; b. 𝒍𝒊𝒎 𝒙→−𝟐 𝒇 𝒙 𝒏 𝒍𝒊𝒎 𝒙→−𝟐 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→−𝟐 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→−𝟐 𝒙 𝟐 + 𝟏𝟔𝒙 + 𝟖 𝟑 = −𝟐 𝟐 + 𝟏𝟔 −𝟐 + 𝟖 𝟑 = 𝟏𝟔 − 𝟑𝟐 + 𝟖 𝟑 = −𝟖 𝟑 = −𝟓𝟏𝟐
  • 20. 2. Evaluate 𝒇 𝒙 = 𝒙 𝟐 − 𝟔𝟒, 𝒈 𝒙 = 𝒙 − 𝟓 a. 𝒍𝒊𝒎 𝒙→𝟗 𝒇 𝒙 𝒈 𝒙 𝐥𝐢𝐦 𝒙→𝟗 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→9 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝟗 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝟗 𝒙 𝟐 − 𝟔𝟒 𝐥𝐢𝐦 𝒙→𝟗 𝐱−𝟓 = 𝟗 𝟐 − 𝟔𝟒 𝟗−𝟓 = 𝟖𝟏 − 𝟔𝟒 𝟒 = 𝟏𝟕 𝟒 = 𝟖𝟑, 𝟓𝟐𝟏
  • 21. 2. Evaluate 𝒇 𝒙 = 𝒙 𝟐 − 𝟔𝟒, 𝒈 𝒙 = 𝒙 − 𝟓 b. 𝒍𝒊𝒎 𝒙→𝟗 𝒈 𝒙 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝟗 𝒈 𝒙 𝒇 𝒙 = 𝐥𝐢𝐦 𝒙→9 𝒈 𝒙 𝐥𝐢𝐦 𝒙→𝟗 𝒇 𝒙 = 𝐥𝐢𝐦 𝒙→𝟗 𝒙 − 𝟓 𝐥𝐢𝐦 𝒙→𝟗 𝐱 𝟐−𝟔𝟒 = (𝟗 − 𝟓) 𝟗 𝟐−𝟔𝟒 = (𝟒) 𝟖𝟏−𝟔𝟒 = 𝟒 𝟏𝟕 = 𝟏𝟕, 𝟏𝟕𝟗, 𝟖𝟔𝟗, 𝟏𝟖𝟒
  • 22. 3. Evaluate 𝑏 = 𝟏𝟓, 𝑓 𝑥 = 𝒙 𝟐 + 𝟐𝒙 + 𝟏 a. 𝐥𝐢𝐦 𝒙→𝟐 𝒃 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→𝟐 𝒃 )𝒇(𝒙 = 𝒍𝒊𝒎 𝒙→𝟐 𝟏𝟓 𝒙 𝟐+𝟐𝒙+𝟏 = 𝒍𝒊𝒎 𝒙→𝟐 𝟏𝟓 𝒍𝒊𝒎 𝒙→𝟐 𝒙 𝟐+𝟐𝒙+𝟏 = 𝟏𝟓 𝒙 𝟐+𝟐𝒙+𝟏 = 𝟏𝟓 𝟐 𝟐+𝟐 𝟐 +𝟏 = 𝟏𝟓 𝟗 = 𝟑𝟖, 𝟒𝟒𝟑, 𝟑𝟓𝟗, 𝟑𝟕𝟓
  • 23. 3. Evaluate 𝑏 = 𝟏𝟓, 𝑓 𝑥 = 𝒙 𝟐 + 𝟐𝒙 + 𝟏 b. 𝐥𝐢𝐦 𝒙→−𝟑 𝒃 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→−𝟑 𝒃 )𝒇(𝒙 = 𝒍𝒊𝒎 𝒙→−𝟑 𝟏𝟓 𝒙 𝟐+𝟐𝒙+𝟏 = 𝒍𝒊𝒎 𝒙→−𝟑 𝟏𝟓 𝒍𝒊𝒎 𝒙→−𝟑 𝒙 𝟐+𝟐𝒙+𝟏 = 𝟏𝟓 𝒙 𝟐+𝟐𝒙+𝟏 = 𝟏𝟓 −𝟑 𝟐+𝟐 −𝟑 +𝟏 = 𝟏𝟓 𝟒 = 𝟓𝟎, 𝟔𝟐𝟓
  • 25. Evaluate 𝒇 𝒙 = 𝟏𝟔𝒙 𝟐 − 𝟖𝒙 + 𝟏, 𝒈 𝒙 = 𝟔𝟒𝒙 𝟐 + 𝟒𝟖𝒙 + 𝟗, 𝒏 = 𝟐; a. 𝒍𝒊𝒎 𝒙→𝟓 𝒏 𝒇(𝒙) b. 𝒍𝒊𝒎 𝒙→−𝟓 𝒇 𝒙 𝒏 c. 𝒍𝒊𝒎 𝒙→𝟖 𝒏 𝒈(𝒙) d. 𝒍𝒊𝒎 𝒙→−𝟖 𝒈 𝒙 𝒏
  • 26. Evaluate 𝒇 𝒙 = 𝟗𝒛 𝟐 − 𝟏𝟎𝟎, 𝒈 𝒙 = 𝒛 𝟐 − 𝟐𝟓 a. 𝒍𝒊𝒎 𝒛→−𝟑 𝒇 𝒙 𝒈 𝒙 b. 𝒍𝒊𝒎 𝒛→𝟑 𝒈 𝒙 𝒇 𝒙 Evaluate 𝒃 = 𝟑𝟔, 𝒇 𝒙 = 𝒙 𝟑 − 𝟐𝟕 a. 𝐥𝐢𝐦 𝒙→𝟑 𝒃 𝒇(𝒙) b. 𝐥𝐢𝐦 𝒙→𝟐 𝒃 𝒇(𝒙)
  • 28. 1. Evaluate 𝒇 𝒙 = 𝟒 𝒙 𝟐 − 𝟔𝒙 + 𝟗, 𝒈 𝒙 = 𝒙 𝟑 + 𝟖𝒙 𝟐 − 𝟏𝟔𝒙 + 𝟒, 𝒏 = 𝟑; a. 𝒍𝒊𝒎 𝒙→𝟑 𝒏 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→𝟑 𝒏 )𝒇(𝒙 = 𝒏 𝒍𝒊𝒎 𝒙→𝟑 )𝒇(𝒙 = 𝟑 𝒍𝒊𝒎 𝒙→𝟑 𝟒 𝒙 𝟐 − 𝟔𝒙 + 𝟗 = 3 𝟒 𝟑 𝟐 − 𝟔 𝟑 + 𝟗 = 3 𝟑𝟔 − 𝟏𝟖 + 𝟗 = 𝟑= 3 𝟐𝟕
  • 29. 1. Evaluate 𝒇 𝒙 = 𝟒 𝒙 𝟐 − 𝟔𝒙 + 𝟗, 𝒈 𝒙 = 𝒙 𝟑 + 𝟖𝒙 𝟐 − 𝟏𝟔𝒙 + 𝟒, 𝒏 = 𝟑; b. 𝒍𝒊𝒎 𝒙→−𝟑 𝒇 𝒙 𝒏 𝒍𝒊𝒎 𝒙→−𝟑 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→−𝟑 𝒇 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→−𝟑 𝟒𝒙 𝟐 − 𝟔𝒙 + 𝟗 𝟑 = 𝟒 −𝟑 𝟐 − 𝟔 −𝟑 + 𝟗 𝟑 = 𝟑𝟔 + 𝟏𝟖 + 𝟗 𝟑 = 𝟔𝟑 𝟑 = 𝟐𝟓𝟎, 𝟎𝟒𝟕
  • 30. 1. Evaluate 𝒇 𝒙 = 𝟒 𝒙 𝟐 − 𝟔𝒙 + 𝟗, 𝒈 𝒙 = 𝒙 𝟑 + 𝟖𝒙 𝟐 − 𝟏𝟔𝒙 + 𝟒, 𝒏 = 𝟑; c. 𝒍𝒊𝒎 𝒙→𝟐 𝒏 𝒈(𝒙) 𝒍𝒊𝒎 𝒙→𝟐 𝒏 )𝒈(𝒙 = 𝒏 𝒍𝒊𝒎 𝒙→𝟐 )𝒈(𝒙 = 𝟑 𝒍𝒊𝒎 𝒙→𝟐 𝒙 𝟑 + 𝟖𝒙 𝟐 − 𝟏𝟔𝒙 + 𝟒 = 3 𝟐 𝟑 + 𝟖 𝟐 𝟐 − 𝟏𝟔 𝟐 + 𝟒 = 3 𝟖 + 𝟑𝟐 − 𝟑𝟐 + 𝟒 = 𝟐. 𝟐𝟗= 3 𝟏𝟐
  • 31. 1. Evaluate 𝒇 𝒙 = 𝟒 𝒙 𝟐 − 𝟔𝒙 + 𝟗, 𝒈 𝒙 = 𝒙 𝟑 + 𝟖𝒙 𝟐 − 𝟏𝟔𝒙 + 𝟒, 𝒏 = 𝟑; d. 𝒍𝒊𝒎 𝒙→−𝟐 𝒈 𝒙 𝒏 𝒍𝒊𝒎 𝒙→−𝟐 𝒈 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→−𝟐 𝒈 𝒙 𝒏 = 𝒍𝒊𝒎 𝒙→−𝟐 𝒙 𝟑 + 𝟖𝒙 𝟐 − 𝟏𝟔𝒙 + 𝟒 𝟑 = −𝟐 𝟑 + 𝟖 −𝟐 𝟐 − 𝟏𝟔 −𝟐 + 𝟒 𝟑 = −𝟖 + 𝟑𝟐 + 𝟑𝟐 + 𝟒 𝟑 = 𝟔𝟎 𝟑 = 𝟐𝟏𝟔, 𝟎𝟎𝟎
  • 32. 2. Evaluate 𝒇 𝒙 = 𝟏𝟔𝐱 𝟐 − 𝟐𝟓, 𝒈 𝒙 = 𝒙 𝟑 − 𝟏 a. 𝒍𝒊𝒎 𝒙→𝟑 𝒇 𝒙 𝒈 𝒙 𝐥𝐢𝐦 𝒙→𝟑 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→3 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝟑 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝟑 𝟏𝟔𝒙 𝟐 − 𝟐𝟓 𝐥𝐢𝐦 𝒙→𝟑 𝐱 𝟑−𝟏 = 𝟏𝟔 𝟑 𝟐 − 𝟐𝟓 𝟑 𝟑−𝟏 = 𝟏𝟒𝟒 − 𝟐𝟓 𝟐𝟔 = 𝟏𝟏𝟗 𝟐𝟔 = 𝟗. 𝟐𝟏 × 𝟏𝟎 𝟓𝟑
  • 33. 2. Evaluate 𝒇 𝒙 = 𝟏𝟔𝐱 𝟐 − 𝟐𝟓, 𝒈 𝒙 = 𝒙 𝟑 − 𝟏 b. 𝒍𝒊𝒎 𝒙→𝟑 𝒈 𝒙 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝟑 𝒈 𝒙 𝒇 𝒙 = 𝐥𝐢𝐦 𝒙→3 𝒈 𝒙 𝐥𝐢𝐦 𝒙→𝟑 𝒇 𝒙 = 𝐥𝐢𝐦 𝒙→𝟑 𝒙 𝟑 − 𝟏 𝐥𝐢𝐦 𝒙→𝟑 𝟏𝟔𝐱 𝟐−𝟐𝟓 = ( 𝟑 𝟑 − 𝟏) 𝟏𝟔(𝟑) 𝟐−𝟐𝟓 = (𝟐𝟔) 𝟏𝟒𝟒−𝟐𝟓 = 𝟐𝟔 𝟏𝟏𝟗 = 𝟐. 𝟒𝟏 × 𝟏𝟎 𝟏𝟔𝟖
  • 34. 3. Evaluate 𝑏 = 𝟐𝟒, 𝑓 𝑥 = 𝒙 𝟑 − 𝟖 a. 𝐥𝐢𝐦 𝒙→𝟑 𝒃 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→𝟑 𝒃 )𝒇(𝒙 = 𝒍𝒊𝒎 𝒙→𝟑 𝟐𝟒 𝒙 𝟑−𝟖 = 𝒍𝒊𝒎 𝒙→𝟑 𝟐𝟒 𝒍𝒊𝒎 𝒙→𝟑 𝒙 𝟑−𝟖 = 𝟐𝟒 𝒙 𝟑−𝟖 = 𝟐𝟒 𝟑 𝟑−𝟖 = 𝟐𝟒 𝟏𝟗 = 𝟏. 𝟔𝟕 × 𝟏𝟎 𝟐𝟔
  • 35. 3. Evaluate 𝑏 = 𝟐𝟒, 𝑓 𝑥 = 𝒙 𝟑 − 𝟖 b. 𝐥𝐢𝐦 𝒙→𝟐 𝒃 𝒇(𝒙) 𝒍𝒊𝒎 𝒙→𝟐 𝒃 )𝒇(𝒙 = 𝒍𝒊𝒎 𝒙→𝟐 𝟐𝟒 𝒙 𝟑−𝟖 = 𝒍𝒊𝒎 𝒙→𝟐 𝟐𝟒 𝒍𝒊𝒎 𝒙→𝟐 𝒙 𝟑−𝟖 = 𝟐𝟒 𝒙 𝟑−𝟖 = 𝟐𝟒 𝟐 𝟑−𝟖 = 𝟐𝟒 𝟎 = 𝟏
  • 36. Limit of Logarithmic Function Let a and b be real numbers, where a > 0, b > 0 and b ≠ 1. Then, 𝒍𝒊𝒎 𝒙→𝒂 ( 𝒍𝒐𝒈 𝒃 𝒙) = 𝒍𝒐𝒈 𝒃 𝒂
  • 37. Properties There are three basic properties in limits, which are used as formulas in evaluating the limits of logarithmic functions. 1.Product Rule 2.Quotient Rule 3.Power Rule
  • 38. 1. Product Rule Example: Solution: lim 𝑥→𝑎 (log 𝑏 𝐴𝐵) = lim 𝑥→𝑎 log 𝑏 𝐴 + log 𝑏 𝐵 Given 𝑨 = 𝒙 𝟐 + 𝟐𝒙 + 𝟏, 𝑩 = 𝒙 − 𝟓, 𝒃 = 𝟓 𝒂𝒏𝒅 𝒙 = 𝟑, find 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) 𝒍𝒊𝒎 𝒙→𝒂 𝐥𝐨𝐠 𝒃 𝑨𝑩 = 𝐥𝐢𝐦 𝒙→𝟑 𝐥𝐨𝐠 𝟓(𝒙 𝟐 +𝟐𝒙 + 𝟏) + 𝐥𝐨𝐠 𝟓 𝒙 − 𝟓) = 𝐥𝐨𝐠 𝟓 𝟑 𝟐 + 𝟐 𝟑 + 𝟏 + 𝟑 − 𝟓 = 𝐥𝐨𝐠 𝟓 𝟗 + 𝟔 + 𝟏 − 𝟐 = 𝐥𝐨𝐠 𝟓 𝟏𝟒 = 𝟏. 𝟔𝟒
  • 39. 2. Quotient Rule Example: Solution: lim 𝑥→𝑎 log 𝑏( 𝐴 𝐵 ) = lim 𝑥→𝑎 log 𝑏 𝐴 − log 𝑏 𝐵 Given 𝑨 = 𝒙 𝟐 + 𝟐𝒙 + 𝟏, 𝑩 = 𝒙 − 𝟓, 𝒃 = 𝟓 𝒂𝒏𝒅 𝒙 = 𝟑, find 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨 𝑩 ) 𝒍𝒊𝒎 𝒙→𝒂 𝐥𝐨𝐠 𝒃 𝑨 𝑩 = 𝐥𝐢𝐦 𝒙→𝟑 𝐥𝐨𝐠 𝟓(𝒙 𝟐 +𝟐𝒙 + 𝟏) − 𝐥𝐨𝐠 𝟓 𝒙 − 𝟓) = 𝐥𝐨𝐠 𝟓 𝟑 𝟐 + 𝟐 𝟑 + 𝟏 − 𝟑 − 𝟓 = 𝐥𝐨𝐠 𝟓 𝟗 + 𝟔 + 𝟏 + 𝟐 = 𝐥𝐨𝐠 𝟓 𝟏𝟖 = 𝟏. 𝟕𝟗𝟔
  • 40. 3. Power Rule Example: Solution: 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏 = 𝒍𝒊𝒎 𝒙→𝒂 𝒏 𝒍𝒐𝒈 𝒃 𝑨 Given 𝑨 = 𝒙 − 𝟓, 𝒏 = 𝟑, 𝒃 = 𝟐 𝒂𝒏𝒅 𝒙 = 𝟖, find 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏 𝒍𝒊𝒎 𝒙→𝟖 𝒍𝒐𝒈 𝟐 𝒙 − 𝟓 𝟑 = 𝒍𝒊𝒎 𝒙→𝟖 𝟑 𝒍𝒐𝒈 𝟐 𝒙 − 𝟓 = 𝟑 𝒍𝒐𝒈 𝟐 𝟖 − 𝟓 = 𝟑 𝒍𝒐𝒈 𝟐 𝟑 = 𝟑 𝟏. 𝟓𝟖𝟓 = 𝟒. 𝟕𝟓𝟓
  • 41. Let’s Practice!!! 1. Given 𝑨 = 𝒙 𝟐 + 𝟒𝒙 + 𝟒, 𝑩 = 𝒙 𝟐 − 𝟏𝟔, 𝒙 = 𝟐, 𝒃 = 𝟏𝟎, 𝒏 = 𝟒 ; Find the following: a. 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏
  • 42. Solution: 𝒍𝒊𝒎 𝒙→𝒂 𝐥𝐨𝐠 𝒃 𝑨𝑩 = 𝐥𝐢𝐦 𝒙→𝟐 𝐥𝐨𝐠 𝟏𝟎(𝒙 𝟐 + 𝟒𝒙 + 𝟒) + 𝐥𝐨𝐠 𝟏𝟎(𝒙 𝟐 − 𝟏𝟔) = 𝐥𝐨𝐠 𝟏𝟎 𝟐 𝟐 + 𝟒 𝟐 + 𝟒 + 𝟐 𝟐 − 𝟏𝟔 = 𝐥𝐨𝐠 𝟏𝟎 𝟒 + 𝟖 + 𝟒 + 𝟒 − 𝟏𝟔 = 𝐥𝐨𝐠 𝟏𝟎 𝟒 = 𝟎. 𝟔𝟎𝟐 1. Given 𝑨 = 𝒙 𝟐 + 𝟒𝒙 + 𝟒, 𝑩 = 𝒙 𝟐 − 𝟏𝟔, 𝒙 = 𝟐, 𝒃 = 𝟏𝟎, 𝒏 = 𝟒 ;
  • 43. Solution: 𝒍𝒊𝒎 𝒙→𝒂 𝐥𝐨𝐠 𝒃 𝑨 𝑩 = 𝐥𝐢𝐦 𝒙→𝟐 𝐥𝐨𝐠 𝟏𝟎 𝒙 𝟐 + 𝟒𝒙 + 𝟒 − 𝐥𝐨𝐠 𝟏𝟎(𝒙 𝟐 − 𝟏𝟔) 1. Given 𝑨 = 𝒙 𝟐 + 𝟒𝒙 + 𝟒, 𝑩 = 𝒙 𝟐 − 𝟏𝟔, 𝒙 = 𝟐, 𝒃 = 𝟏𝟎, 𝒏 = 𝟒 ; = 𝐥𝐨𝐠 𝟏𝟎 𝟐 𝟐 + 𝟒 𝟐 + 𝟒 − 𝟐 𝟐 − 𝟏𝟔 = 𝐥𝐨𝐠 𝟏𝟎 𝟒 + 𝟖 + 𝟒 − 𝟒 + 𝟏𝟔 = 𝐥𝐨𝐠 𝟏𝟎 𝟐𝟖 = 𝟏. 𝟒𝟒𝟕
  • 44. Solution: 𝒍𝒊𝒎 𝒙→𝟐 𝒍𝒐𝒈 𝟏𝟎 𝒙 𝟐 + 𝟒𝒙 + 𝟒 𝟒 = 𝒍𝒊𝒎 𝒙→𝟐 𝟒 𝒍𝒐𝒈 𝟏𝟎 𝒙 𝟐 + 𝟒𝒙 + 𝟒 = 𝟒 𝒍𝒐𝒈 𝟏𝟎 𝟐 𝟐 + 𝟒 𝟐 + 𝟒 = 𝟒 𝒍𝒐𝒈 𝟏𝟎 𝟏𝟔 = 𝟒 𝟏. 𝟐𝟎𝟒 = 𝟒. 𝟖𝟏𝟔 1. Given 𝑨 = 𝒙 𝟐 + 𝟒𝒙 + 𝟒, 𝑩 = 𝒙 𝟐 − 𝟏𝟔, 𝒙 = 𝟐, 𝒃 = 𝟏𝟎, 𝒏 = 𝟒 ;
  • 46. 1. Given 𝑨 = 𝟗𝒙 𝟐 + 𝟑𝟎𝒙 + 𝟐𝟓, 𝑩 = 𝟒𝒙 𝟐 − 𝟏𝟔, 𝒙 = 𝟓, 𝒃 = 𝟑𝟓, 𝒏 = 𝟏𝟓 ; 2. Given 𝑨 = 𝒙 𝟐 − 𝟏𝟔𝒙 + 𝟔𝟒, 𝑩 = 𝒙 𝟐 − 𝟏, 𝒙 = 𝟑, 𝒃 = 𝟐𝟓, 𝒏 = 𝟐𝟓 Find the following: a. 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏
  • 47. 2. Given 𝑨 = 𝒙 𝟐 + 𝟐𝟎𝒙 − 𝟏𝟎𝟎, 𝑩 = 𝒙 𝟐 − 𝟑𝟔, 𝒙 = 𝟏𝟐, 𝒃 = 𝟑𝟔, 𝒏 = 𝟐𝟓 ; Find the following: a. 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏
  • 49. Solution: 𝒍𝒊𝒎 𝒙→𝒂 𝐥𝐨𝐠 𝒃 𝑨𝑩 = 𝐥𝐢𝐦 𝒙→𝟐 𝐥𝐨𝐠 𝟏𝟓(𝟐𝟓𝒙 𝟐 + 𝟒𝟎𝒙 + 𝟏𝟔) + 𝐥𝐨𝐠 𝟏𝟓(𝒙 𝟐 − 𝟔𝟒) = 𝐥𝐨𝐠 𝟏𝟓 𝟐𝟓 𝟐 𝟐 + 𝟒𝟎 𝟐 + 𝟏𝟔 + 𝟐 𝟐 − 𝟔𝟒 = 𝐥𝐨𝐠 𝟏𝟓 𝟏𝟎𝟎 + 𝟖𝟎 + 𝟏𝟔 + 𝟒 − 𝟔𝟒 = 𝐥𝐨𝐠 𝟏𝟓 𝟏𝟑𝟔 = 𝟏. 𝟖𝟏𝟒 1. Given 𝑨 = 𝟐𝟓𝒙 𝟐 + 𝟒𝟎𝒙 + 𝟏𝟔, 𝑩 = 𝒙 𝟐 − 𝟔𝟒, 𝒙 = 𝟐, 𝒃 = 𝟏𝟓, 𝒏 = 𝟓 ;
  • 50. Solution: 𝒍𝒊𝒎 𝒙→𝒂 𝐥𝐨𝐠 𝒃 𝑨 𝑩 = 𝐥𝐢𝐦 𝒙→𝟐 𝐥𝐨𝐠 𝟏𝟓 𝟐𝟓𝒙 𝟐 + 𝟒𝟎𝒙 + 𝟏𝟔 − 𝐥𝐨𝐠 𝟏𝟓(𝒙 𝟐 − 𝟔𝟒) 1. Given 𝑨 = 𝟐𝟓𝒙 𝟐 + 𝟒𝟎𝒙 + 𝟏𝟔, 𝑩 = 𝒙 𝟐 − 𝟔𝟒, 𝒙 = 𝟐, 𝒃 = 𝟏𝟓, 𝒏 = 𝟓 ; = 𝐥𝐨𝐠 𝟏𝟓 𝟐𝟓 𝟐 𝟐 + 𝟒𝟎 𝟐 + 𝟏𝟔 − 𝟐 𝟐 − 𝟔𝟒 = 𝐥𝐨𝐠 𝟏𝟓 𝟏𝟎𝟎 + 𝟖𝟎 + 𝟏𝟔 − 𝟒 + 𝟔𝟒 = 𝐥𝐨𝐠 𝟏𝟓 𝟐𝟓𝟔 = 𝟐. 𝟎𝟒𝟖
  • 51. Solution: 𝒍𝒊𝒎 𝒙→𝟐 𝒍𝒐𝒈 𝟏𝟓 𝟐𝟓𝒙 𝟐 + 𝟒𝟎𝒙 + 𝟏𝟔 𝟓 = 𝒍𝒊𝒎 𝒙→𝟐 𝟓 𝒍𝒐𝒈 𝟏𝟓 𝟐𝟓𝒙 𝟐 + 𝟒𝟎𝒙 + 𝟏𝟔 = 𝟓 𝒍𝒐𝒈 𝟏𝟓 𝟐𝟓 𝟐 𝟐 + 𝟒𝟎 𝟐 + 𝟏𝟔 = 𝟓 𝒍𝒐𝒈 𝟏𝟓 𝟏𝟗𝟔 = 𝟓 𝟏. 𝟗𝟒𝟗 = 𝟗. 𝟕𝟒𝟓 1. Given 𝑨 = 𝟐𝟓𝒙 𝟐 + 𝟒𝟎𝒙 + 𝟏𝟔, 𝑩 = 𝒙 𝟐 − 𝟔𝟒, 𝒙 = 𝟐, 𝒃 = 𝟏𝟓, 𝒏 = 𝟓 ;
  • 52. Solution: 𝒍𝒊𝒎 𝒙→𝒂 𝐥𝐨𝐠 𝒃 𝑨𝑩 = 𝐥𝐢𝐦 𝒙→𝟏𝟐 𝐥𝐨𝐠 𝟑𝟔(𝒙 𝟐 + 𝟐𝟎𝒙 − 𝟏𝟎𝟎) + 𝐥𝐨𝐠 𝟑𝟔(𝒙 𝟐 − 𝟑𝟔) = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟐 𝟐 + 𝟐𝟎 𝟏𝟐 − 𝟏𝟎𝟎 + 𝟏𝟐 𝟐 − 𝟑𝟔 = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟒𝟒 + 𝟐𝟒𝟎 − 𝟏𝟎𝟎 + 𝟏𝟒𝟒 − 𝟑𝟔 = 𝐥𝐨𝐠 𝟑𝟔 𝟑𝟗𝟐 = 𝟏. 𝟔𝟔𝟔 2. Given 𝑨 = 𝒙 𝟐 + 𝟐𝟎𝒙 − 𝟏𝟎𝟎, 𝑩 = 𝒙 𝟐 − 𝟑𝟔, 𝒙 = 𝟏𝟐, 𝒃 = 𝟑𝟔, 𝒏 = 𝟐𝟓 ;
  • 53. Solution: 𝒍𝒊𝒎 𝒙→𝒂 𝐥𝐨𝐠 𝒃 𝑨 𝑩 = 𝐥𝐢𝐦 𝒙→𝟏𝟐 𝐥𝐨𝐠 𝟑𝟔 𝒙 𝟐 + 𝟐𝟎𝒙 − 𝟏𝟎𝟎 − 𝐥𝐨𝐠 𝟑𝟔(𝒙 𝟐 − 𝟑𝟔) 2. Given 𝑨 = 𝒙 𝟐 + 𝟐𝟎𝒙 − 𝟏𝟎𝟎, 𝑩 = 𝒙 𝟐 − 𝟑𝟔, 𝒙 = 𝟏𝟐, 𝒃 = 𝟑𝟔, 𝒏 = 𝟐𝟓 ; = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟐 𝟐 + 𝟐𝟎 𝟏𝟐 − 𝟏𝟎𝟎 − 𝟏𝟐 𝟐 − 𝟑𝟔 = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟒𝟒 + 𝟐𝟒𝟎 − 𝟏𝟎𝟎 − 𝟏𝟒𝟒 + 𝟑𝟔 = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟕𝟔 = 𝟏. 𝟒𝟒𝟑
  • 54. Solution: 𝒍𝒊𝒎 𝒙→𝟏𝟐 𝒍𝒐𝒈 𝟑𝟔 𝒙 𝟐 + 𝟐𝟎𝒙 − 𝟏𝟎𝟎 𝟐𝟓 = 𝒍𝒊𝒎 𝒙→𝟏𝟐 𝟐𝟓 𝒍𝒐𝒈 𝟑𝟔 𝒙 𝟐 + 𝟐𝟎𝒙 − 𝟏𝟎𝟎 = 𝟐𝟓 𝒍𝒐𝒈 𝟑𝟔 𝟏𝟐 𝟐 + 𝟐𝟎 𝟏𝟐 − 𝟏𝟎𝟎 = 𝟐𝟓 𝒍𝒐𝒈 𝟑𝟔 𝟐𝟖𝟒 = 𝟐𝟓 𝟏. 𝟓𝟕𝟔 = 𝟑𝟗. 𝟒𝟎𝟗 2. Given 𝑨 = 𝒙 𝟐 + 𝟐𝟎𝒙 − 𝟏𝟎𝟎, 𝑩 = 𝒙 𝟐 − 𝟑𝟔, 𝒙 = 𝟏𝟐, 𝒃 = 𝟑𝟔, 𝒏 = 𝟐𝟓 ;
  • 56. Given 𝑨 = 𝟑𝟔𝒚 𝟐 − 𝟗𝟔𝒚 + 𝟔𝟒, 𝑩 = 𝒚 𝟐 − 𝟔𝟒, 𝒚 = 𝟓, 𝒃 = 𝟐𝟓, 𝒏 = 𝟖 ; Find the following: a. 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏
  • 57. Given 𝑨 = 𝟒𝟓𝒎 𝟐 − 𝟑𝟔𝒎 − 𝟗, 𝑩 = 𝒎 𝟐 − 𝟐𝒎 + 𝟐, 𝒎 = 𝟔, 𝒃 = 𝟑𝟔, 𝒏 = 𝟓 ; Find the following: a. 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏
  • 58. Given 𝑨 = 𝟏𝟔𝒚 𝟐 − 𝟒𝟖𝒚 + 𝟑𝟔, 𝑩 = 𝒚 𝟑 − 𝟏, 𝒚 = 𝟑, 𝒃 = 𝟏𝟖, 𝒏 = 𝟔 ; Find the following: a. 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏
  • 60. Solution: 𝒍𝒊𝒎 𝐲→𝒂 𝐥𝐨𝐠 𝒃 𝑨𝑩 = 𝐥𝐢𝐦 𝐲→𝟓 𝐥𝐨𝐠 𝟐𝟓(𝟑𝟔𝒚 𝟐 − 𝟗𝟔𝒚 + 𝟔𝟒) + 𝐥𝐨𝐠 𝟐𝟓(𝒚 𝟐 − 𝟔𝟒) = 𝐥𝐨𝐠 𝟐𝟓 𝟑𝟔 𝟓 𝟐 − 𝟗𝟔 𝟓 + 𝟔𝟒 + 𝟓 𝟐 − 𝟔𝟒 = 𝐥𝐨𝐠 𝟐𝟓 𝟗𝟎𝟎 − 𝟒𝟖𝟎 + 𝟔𝟒 + 𝟐𝟓 − 𝟔𝟒 = 𝐥𝐨𝐠 𝟐𝟓 𝟒𝟒𝟓 = 𝟏. 𝟖𝟗𝟒 1. Given 𝑨 = 𝟑𝟔𝒚 𝟐 − 𝟗𝟔𝒚 + 𝟔𝟒, 𝑩 = 𝒚 𝟐 − 𝟔𝟒, 𝒚 = 𝟓, 𝒃 = 𝟐𝟓, 𝒏 = 𝟖
  • 61. Solution: = 𝟏. 𝟗𝟒𝟓 1. Given 𝑨 = 𝟑𝟔𝒚 𝟐 − 𝟗𝟔𝒚 + 𝟔𝟒, 𝑩 = 𝒚 𝟐 − 𝟔𝟒, 𝒚 = 𝟓, 𝒃 = 𝟐𝟓, 𝒏 = 𝟖 𝒍𝒊𝒎 𝐲→𝒂 𝐥𝐨𝐠 𝒃 𝑨 𝑩 = 𝐥𝐢𝐦 𝐲→𝟓 𝐥𝐨𝐠 𝟐𝟓 𝟑𝟔𝒚 𝟐 − 𝟗𝟔𝒚 + 𝟔𝟒 − 𝐥𝐨𝐠 𝟐𝟓(𝒚 𝟐 − 𝟔𝟒) = 𝐥𝐨𝐠 𝟐𝟓 𝟑𝟔 𝟓 𝟐 − 𝟗𝟔 𝟓 + 𝟔𝟒 − 𝟓 𝟐 − 𝟔𝟒 = 𝐥𝐨𝐠 𝟐𝟓 𝟗𝟎𝟎 − 𝟒𝟖𝟎 + 𝟔𝟒 − 𝟐𝟓 + 𝟔𝟒 = 𝐥𝐨𝐠 𝟐𝟓 𝟓𝟐𝟑
  • 62. Solution: 𝒍𝒊𝒎 𝒚→𝟓 𝒍𝒐𝒈 𝟐𝟓 𝟑𝟔𝒚 𝟐 − 𝟗𝟔𝒚 + 𝟔𝟒 𝟖 = 𝒍𝒊𝒎 𝒚→𝟓 𝟖 𝒍𝒐𝒈 𝟐𝟓 𝟑𝟔𝒚 𝟐 − 𝟗𝟔𝒚 + 𝟔𝟒 = 𝟖 𝒍𝒐𝒈 𝟐𝟓 𝟑𝟔 𝟓 𝟐 − 𝟗𝟔 𝟓 + +𝟔𝟒 = 𝟖 𝒍𝒐𝒈 𝟐𝟓 𝟒𝟖𝟒 = 𝟖 𝟏. 𝟗𝟐𝟏 = 𝟏𝟓. 𝟑𝟔𝟓 1. Given 𝑨 = 𝟑𝟔𝒚 𝟐 − 𝟗𝟔𝒚 + 𝟔𝟒, 𝑩 = 𝒚 𝟐 − 𝟔𝟒, 𝒚 = 𝟓, 𝒃 = 𝟐𝟓, 𝒏 = 𝟖
  • 63. Solution: 𝒍𝒊𝒎 𝐦→𝒂 𝐥𝐨𝐠 𝒃 𝑨𝑩 = 𝐥𝐢𝐦 𝐦→𝟔 𝐥𝐨𝐠 𝟑𝟔(𝟒𝟓𝒎 𝟐 − 𝟑𝟔𝒎 − 𝟗) + 𝐥𝐨𝐠 𝟑𝟔(𝒎 𝟐 − 𝟐𝒎 + 𝟐) = 𝐥𝐨𝐠 𝟑𝟔 𝟒𝟓 𝟔 𝟐 − 𝟑𝟔 𝟔 − 𝟗 + 𝟔 𝟐 − 𝟐 𝟔 + 𝟐 = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟔𝟐𝟎 − 𝟐𝟏𝟔 − 𝟗 + 𝟑𝟔 − 𝟏𝟐 + 𝟐 = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟒𝟐𝟏 = 𝟐. 𝟎𝟐𝟔 2. Given 𝑨 = 𝟒𝟓𝒎 𝟐 − 𝟑𝟔𝒎 − 𝟗, 𝑩 = 𝒎 𝟐 − 𝟐𝒎 + 𝟐, 𝒎 = 𝟔, 𝒃 = 𝟑𝟔, 𝒏 = 𝟓 ;
  • 64. Solution: = 𝟐. 𝟎𝟏𝟓 2. Given 𝑨 = 𝟒𝟓𝒎 𝟐 − 𝟑𝟔𝒎 − 𝟗, 𝑩 = 𝒎 𝟐 − 𝟐𝒎 + 𝟐, 𝒎 = 𝟔, 𝒃 = 𝟑𝟔, 𝒏 = 𝟓 ; 𝒍𝒊𝒎 𝐦→𝒂 𝐥𝐨𝐠 𝒃 𝑨 𝑩 = 𝐥𝐢𝐦 𝐦→𝟔 𝐥𝐨𝐠 𝟑𝟔 𝟒𝟓𝒎 𝟐 − 𝟑𝟔𝒎 − 𝟗 − 𝐥𝐨𝐠 𝟑𝟔(𝒎 𝟐 − 𝟐𝒎 + 𝟐) = 𝐥𝐨𝐠 𝟑𝟔 𝟒𝟓 𝟔 𝟐 − 𝟑𝟔 𝟔 − 𝟗 − 𝟔 𝟐 − 𝟐 𝟔 + 𝟐 = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟔𝟐𝟎 − 𝟐𝟏𝟔 − 𝟗 − 𝟑𝟔 + 𝟏𝟐 − 𝟐 = 𝐥𝐨𝐠 𝟑𝟔 𝟏𝟑𝟔𝟗
  • 65. Solution: 𝒍𝒊𝒎 𝒎→𝟔 𝒍𝒐𝒈 𝟑𝟔 𝟒𝟓𝒎 𝟐 − 𝟑𝟔𝒎 − 𝟗 𝟓 = 𝒍𝒊𝒎 𝒙→𝟐 𝟓 𝒍𝒐𝒈 𝟑𝟔 𝟒𝟓𝒎 𝟐 − 𝟑𝟔𝒎 − 𝟗 = 𝟓 𝒍𝒐𝒈 𝟑𝟔 𝟒𝟓 𝟔 𝟐 − 𝟑𝟔 𝟔 − 𝟗 = 𝟓 𝒍𝒐𝒈 𝟑𝟔 𝟏𝟑𝟗𝟓 = 𝟓 𝟐. 𝟎𝟐𝟏 = 𝟏𝟎. 𝟏𝟎𝟑 2. Given 𝑨 = 𝟒𝟓𝒎 𝟐 − 𝟑𝟔𝒎 − 𝟗, 𝑩 = 𝒎 𝟐 − 𝟐𝒎 + 𝟐, 𝒎 = 𝟔, 𝒃 = 𝟑𝟔, 𝒏 = 𝟓 ;
  • 66. Solution: 𝒍𝒊𝒎 𝐲→𝒂 𝐥𝐨𝐠 𝒃 𝑨𝑩 = 𝐥𝐢𝐦 𝒚→𝟑 𝐥𝐨𝐠 𝟏𝟖(𝟏𝟔𝒚 𝟐 − 𝟒𝟔𝒚 + 𝟑𝟔) + 𝐥𝐨𝐠 𝟏𝟖(𝒚 𝟑 − 𝟏) = 𝐥𝐨𝐠 𝟏𝟖 𝟏𝟔 𝟑 𝟐 − 𝟒𝟖 𝟑 + 𝟑𝟔 + 𝟑 𝟑 − 𝟏 = 𝐥𝐨𝐠 𝟏𝟖 𝟏𝟒𝟒 − 𝟏𝟒𝟒 + 𝟑𝟔 + 𝟐𝟕 − 𝟏 = 𝐥𝐨𝐠 𝟏𝟖 𝟔𝟐 = 𝟏. 𝟒𝟐𝟖 3. Given 𝑨 = 𝟏𝟔𝒚 𝟐 − 𝟒𝟖𝒚 + 𝟑𝟔, 𝑩 = 𝒚 𝟑 − 𝟏, 𝒚 = 𝟑, 𝒃 = 𝟏𝟖, 𝒏 = 𝟔 ;
  • 67. Solution: = 𝟎. 𝟕𝟗𝟕 3. Given 𝑨 = 𝟏𝟔𝒚 𝟐 − 𝟒𝟖𝒚 + 𝟑𝟔, 𝑩 = 𝒚 𝟑 − 𝟏, 𝒚 = 𝟑, 𝒃 = 𝟏𝟖, 𝒏 = 𝟔 ; 𝒍𝒊𝒎 𝐲→𝒂 𝐥𝐨𝐠 𝒃 𝑨 𝑩 = 𝐥𝐢𝐦 𝒚→𝟑 𝐥𝐨𝐠 𝟏𝟖 𝟏𝟔𝒚 𝟐 − 𝟒𝟔𝒚 + 𝟑𝟔 − 𝐥𝐨𝐠 𝟏𝟖(𝒚 𝟑 − 𝟏) = 𝐥𝐨𝐠 𝟏𝟖 𝟏𝟔 𝟑 𝟐 − 𝟒𝟖 𝟑 + 𝟑𝟔 − 𝟑 𝟐 − 𝟏 = 𝐥𝐨𝐠 𝟏𝟖 𝟏𝟒𝟒 − 𝟏𝟒𝟒 + 𝟑𝟔 − 𝟐𝟕 + 𝟏 = 𝐥𝐨𝐠 𝟏𝟖 𝟏𝟎
  • 68. Solution: 𝒍𝒊𝒎 𝒚→𝟑 𝒍𝒐𝒈 𝟏𝟖 𝟏𝟔𝒚 𝟐 − 𝟒𝟖𝒚 + 𝟑𝟔 𝟔 = 𝒍𝒊𝒎 𝒚→𝟑 𝟔 𝒍𝒐𝒈 𝟏𝟖 𝟏𝟔𝒚 𝟐 − 𝟒𝟖𝒚 + 𝟑𝟔 = 𝟔 𝒍𝒐𝒈 𝟏𝟖 𝟏𝟔 𝟑 𝟐 − 𝟒𝟖 𝟑 + 𝟑𝟔 = 𝟔 𝒍𝒐𝒈 𝟏𝟖 𝟑𝟔 = 𝟔 𝟏. 𝟐𝟒𝟎 = 𝟕. 𝟒𝟑𝟗 3. Given 𝑨 = 𝟏𝟔𝒚 𝟐 − 𝟒𝟖𝒚 + 𝟑𝟔, 𝑩 = 𝒚 𝟑 − 𝟏, 𝒚 = 𝟑, 𝒃 = 𝟏𝟖, 𝒏 = 𝟔 ;
  • 69. Pointers to Review Limits of Non – Algebraic Functions Limit of an Exponential Function Limit of Logarithmic Function
  • 70. Given 𝑨 = 𝒚 𝟐 + 𝟔𝒚 + 𝟗, 𝑩 = 𝟒𝒚 𝟐 − 𝟏, 𝒚 = 𝟖, 𝒃 = 𝟏𝟓, 𝒏 = 𝟔 ; Find the following: a. 𝒍𝒊𝒎 𝒚→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒚→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒚→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏
  • 71. Given 𝑨 = 𝟑𝒙 𝟐 + 𝟗𝒙 + 𝟗, 𝑩 = 𝒙 − 𝟓, 𝒙 = 𝟐, 𝒃 = 𝟏𝟑, 𝒏 = 𝟏𝟎 ; Find the following: a. 𝒍𝒊𝒎 𝒙→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒙→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏
  • 72. Given 𝑨 = 𝟔𝒛 𝟐 − 𝟒𝒛 + 𝟓, 𝑩 = 𝟐𝒛 𝟐 + 𝟒𝒛 − 𝟑, 𝒛 = 𝟑, 𝒃 = 𝟖, 𝒏 = 𝟐; Find the following: a. 𝒍𝒊𝒎 𝒛→𝒂 (𝒍𝒐𝒈 𝒃 𝑨𝑩) b. 𝒍𝒊𝒎 𝒛→𝒂 𝒍𝒐𝒈 𝒃( 𝑨 𝑩 ) c. 𝒍𝒊𝒎 𝒛→𝒂 𝒍𝒐𝒈 𝒃 𝑨 𝒏