Chapter 8.1 Limits PDF
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Kolej Matrikulasi Kedah
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This document provides a comprehensive explanation of limits in mathematics. It covers various concepts such as properties of limits, one-sided limits, and infinite limits using examples and detailed definitions. This is a course handout or notes.
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CHAPTER 8.1 LIMITS PROPERTIES OF LIMITS If 𝑓 𝑥 = 𝑘, where k is a constant, thus lim 𝑓(𝑥) = lim 𝑘 = 𝑘 𝑥→𝑎 𝑥→𝑎 If 𝑓 𝑥 = 𝑥 𝑛 , thus lim 𝑓(𝑥) = lim 𝑥 𝑛 = 𝑎𝑛 𝑥→𝑎 𝑥→𝑎 Se...
CHAPTER 8.1 LIMITS PROPERTIES OF LIMITS If 𝑓 𝑥 = 𝑘, where k is a constant, thus lim 𝑓(𝑥) = lim 𝑘 = 𝑘 𝑥→𝑎 𝑥→𝑎 If 𝑓 𝑥 = 𝑥 𝑛 , thus lim 𝑓(𝑥) = lim 𝑥 𝑛 = 𝑎𝑛 𝑥→𝑎 𝑥→𝑎 Senang faham! See x substitute x PROPERTIES OF LIMITS lim 𝑘𝑓(𝑥) = 𝑘 lim 𝑓(𝑥) 𝑥→𝑎 𝑥→𝑎 lim 𝑓 𝑥 ± 𝑔 𝑥 = lim 𝑓(𝑥) ± lim 𝑔(𝑥) 𝑥→𝑎 𝑥→𝑎 𝑥→𝑎 lim 𝑓 𝑥 𝑔 𝑥 = lim 𝑓(𝑥) lim 𝑔(𝑥) 𝑥→𝑎 𝑥→𝑎 𝑥→𝑎 𝑓 𝑥 lim 𝑓 𝑥 𝑥→𝑎 lim = , lim 𝑔(𝑥) ≠ 0 𝑥→𝑎 𝑔 𝑥 lim 𝑔(𝑥) 𝑥→𝑎 𝑥→𝑎 𝟎 CASE (INDETERMINATE) 𝟎 Factorization Multiply with conjugates Cara ini digunakan apabila ADA SURD ONE SIDED LIMIT x approaches from the left x approaches from the right Example: Example: 0- = ‒0.000000…01 0+ = 0.000000…01 1- = 0.999999…99 1+ = 1.000000…01 DOES THE LIMIT EXIST? How To Determine? lim− 𝑓(𝑥) = lim+ 𝑓(𝑥) = 𝐿 𝑥→𝑎 𝑥→𝑎 lim 𝑓(𝑥) = 𝐿 𝑥→𝑎 Tips!! Untuk piecewise function, kalau nak cari a-, guna function yang less than a; kalau nak cari a+, guna function yang more than a INFINITE LIMITS 𝑦 𝑦 = +∞, y > 0 = −∞, y < 0 0+ 0+ 𝑦 𝑦 = −∞, y > 0 = +∞, y < 0 0− 0− LIMITS AT INFINITY 𝑦 =0 ±∞ ∞ CASE (INDETERMINATE) ∞ Divide each term by highest power of x of the denominator 1 1 𝑥 𝑥 = 𝑥2 lim = lim Since 𝑥 → −∞ 𝑥→−∞ 𝑥2 + 1 𝑥→−∞ 𝑥2 1 2 + 2 𝑥 = − 𝑥2 𝑥 𝑥 ⋮
