Limit of a Function, Using Algebraic Simplification

Limit of a Function, Using Algebraic Simplification Properties of Limits: If L, M, c, and k are real numbers and if lim 𝑓(𝑥) = 𝐿 and 𝑥→𝑐 1. 2. lim 𝑔(𝑥) = 𝑀 , then 𝑥→𝑐 lim(𝑓(𝑥) ± 𝑔(𝑥)) = 𝐿 ± 𝑀 : Sum Rule 𝑥→𝑐 lim (𝑓(𝑥) ∙ 𝑔(𝑥)) = 𝐿 ∙ 𝑀 𝑥→𝑐 3. lim (𝑘 ∙ 𝑓(𝑥)) = 𝑘 ∙ 𝐿 𝑥→𝑐 𝑓(𝑥) 4. lim 𝑥→𝑐 𝑔(𝑥) = 𝐿 𝑀 , 𝑀≠0 : Product Rule : Constant Multiple Rule : Quotient Rule 5. lim (𝑓(𝑥))𝑛 = 𝐿𝑛 , 𝑛 a positive integer : Power Rule 𝑥→𝑐 𝑛 6. lim 𝑛√𝑓(𝑥) = √𝐿 = 𝐿1⁄𝑛 , 𝑛 a positive integer: nth Root Rule 𝑥→𝑐 2.6: Limit of a Function at infinity 1%1=0 a÷•÷÷ Note: All properties of limits are valid for limits as 𝑥 → ∞ 𝑜𝑟 as 𝑥 → −∞. Also, for a positive integer 𝑛, lim 1 lim 1 𝑥→−∞ 𝑥 𝑛 = 0, provided that 1 𝑥𝑛 = 0, and ☐ 𝑥→∞ 𝑥 𝑛 is defined. We use the following method to evaluate limit of a rational function at ∞: Divide the numerator and denominator of the expression by 𝑥 𝑛 , where 𝑛 is the highest power in the denominator of the expression. Evaluate the following limit if it exists. :¥:÷¥=É÷(¥¥÷÷ ¥ :÷i:÷:→=÷ 3𝑥+2 lim 1. 𝑥−5 𝑥→∞ = tin = Degmm.=Degden# lim 2. 𝑥→∞ 25,3¥ ÷ ✓ 2−3𝑥 𝑥−5 = →¥¥: - . leni = ③ ( -3 t =¥:f÷1 2 - ¥) ? =D - = Def lim 3. 𝑥→∞ ✗ mum = ÷ 2 𝑥 2 −𝑥−2 ÷☐¥÷ § = =3 . .¥¥÷¥=÷¥¥÷1 𝑥+3 lim 2𝑥 3 +1 𝑥→∞ =÷(÷÷÷→ . Def 5. mum 𝑥 4 +1 > Deg den = . § = - lim 𝑥→∞ 𝑥 3 −1 :¥ ¥ . = . 3𝑥 2 +1 Degnum.ci#den 4. Deg den . ¥ -¥:¥¥÷= = : • / Note: Rules to find the limit at infinity of a rational Function, 𝑓(𝑥) = 1. If Deg 𝑝(𝑥) < Deg 𝑞(𝑥): Ari ✗ 2. If Deg 𝑝(𝑥) = Deg 𝑞(𝑥): Uni ✗→ • +4¥ = , ✗→ A 𝑞(𝑥) : = o → a coefficient of the numerator YId.%aa.im#enmnatn 3. If Deg 𝑝(𝑥) > Deg 𝑞(𝑥): hi 𝑝(𝑥) P¥✗ , = a 6. 7. 8. lim 𝑥→∞ lim 𝑥→∞ lim ( 3𝑥 2 +8𝑥−4 2𝑥 2 +4𝑥−5 ) 2𝑥 2 −1 = = 𝑥 3 +𝑥 2 +1 𝑥 4 +1 𝑥→−∞ 𝑥 3 −1 i = -00 E) 0