Exercicios Calculo (71)

Prévia do material em texto

82 
 
 =
lim
𝑥→−∞
𝑏
𝑥
lim
𝑥→−∞
1 − lim
𝑥→−∞
3
𝑥 + lim
𝑥→−∞
1
𝑥2
− lim
𝑥→−∞
3
𝑥3
 
 =
0
1 − 0 + 0 − 0
 
 = 0. 
 
lim
𝑥→−∞
ℎ(𝑥) = lim
𝑥→−∞
𝑎. 𝑥2
𝑥3 − 3𝑥2 +𝑥 − 3
 
 = lim
𝑥→−∞
𝑎
𝑥
1 −
3
𝑥 +
1
𝑥2
−
3
𝑥3
 
 =
lim
𝑥→−∞
𝑎
𝑥
lim
𝑥→−∞
1 − lim
𝑥→−∞
3
𝑥 + lim
𝑥→−∞
1
𝑥2
− lim
𝑥→−∞
3
𝑥3
 
 =
0
1 − 0 + 0 − 0
 
 = 0. 
 
𝐶𝑜𝑚𝑜 𝑓(𝑥) ≤ 𝑔(𝑥) ≤ ℎ(𝑥) 𝑝𝑎𝑟𝑎 𝑞𝑢𝑎𝑙𝑞𝑢𝑒𝑟 𝑥 0. 
𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑒 𝑎 𝑓𝑢𝑛çã𝑜 𝑓(𝑥) = 2𝑥 −
10
𝑥
, 𝑑𝑒𝑓𝑖𝑛𝑖𝑑𝑎 𝑝𝑒𝑙𝑎 𝑑𝑖𝑓𝑒𝑟𝑒𝑛ç𝑎 𝑒𝑛𝑡𝑟𝑒 𝑓𝑢𝑛çõ𝑒𝑠 
𝑐𝑜𝑛𝑡í𝑛𝑢𝑎𝑠 𝑒𝑚 𝑡𝑜𝑑𝑜𝑠 𝑜𝑠 𝑛ú𝑚𝑒𝑟𝑜𝑠 𝑑𝑜 𝑠𝑒𝑢𝑠 𝑑𝑜𝑚í𝑛𝑖𝑜𝑠 𝑒, 𝑝𝑜𝑟𝑡𝑎𝑛𝑡𝑜, 𝑓 é 𝑐𝑜𝑛𝑡í𝑛𝑢𝑎 
𝑒𝑚 𝑠𝑒𝑢 𝑑𝑜𝑚í𝑛𝑖𝑜, 𝑖𝑠𝑡𝑜 é,𝑥 ∈ ℝ 𝑡𝑎𝑙 𝑞𝑢𝑒 𝑥 ≠ 0. 
 
𝑓(1) = 2 − 10 = −8 𝑒 𝑓(5) = 32 − 2 = 30 
 
𝐶𝑜𝑚𝑜 𝑓 é 𝑐𝑜𝑛𝑡í𝑛𝑢𝑎 𝑒𝑚 ℝ− {0}, 𝑒𝑛𝑡ã𝑜 𝑓 é 𝑐𝑜𝑛𝑡í𝑛𝑢𝑎 𝑛𝑜 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑜 𝑓𝑒𝑐ℎ𝑎𝑑𝑜 [1,5] 𝑒 
0 é 𝑢𝑚 𝑛ú𝑚𝑒𝑟𝑜 𝑒𝑛𝑡𝑟𝑒 𝑓(1) 𝑒 𝑓(5).𝐿𝑜𝑔𝑜,𝑝𝑒𝑙𝑜 𝑇𝑒𝑜𝑟𝑒𝑚𝑎 𝑑𝑜 𝑉𝑎𝑙𝑜𝑟 𝐼𝑛𝑡𝑒𝑟𝑚𝑒𝑑𝑖á𝑟𝑖𝑜, 
𝑒𝑥𝑖𝑠𝑡𝑒 𝑎𝑙𝑔𝑢𝑚 𝑛ú𝑚𝑒𝑟𝑜 𝑥 ∈ (1,5), 𝑥 > 0,𝑡𝑎𝑙 𝑞𝑢𝑒 𝑓(𝑥) = 0.𝑂𝑢 𝑠𝑒𝑗𝑎, 
𝑓(𝑥) = 0⟹ 2𝑥 −
10
𝑥
= 0 ∴ 2𝑥 =
10
𝑥
, 𝑝𝑎𝑟𝑎 𝑎𝑙𝑔𝑢𝑚 𝑥 > 0 𝑒 𝑥 ∈ (1,5). 
 
𝑏) 𝑈𝑠𝑎𝑛𝑑𝑜 𝑎 𝑑𝑒𝑓𝑖𝑛𝑖çã𝑜 𝑑𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑑𝑎 𝑑𝑒 𝑢𝑚𝑎 𝑓𝑢𝑛çã𝑜, 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑎 𝑑𝑒𝑟𝑖𝑣𝑎𝑑𝑎 𝑑𝑒 
𝑓(𝑥) =
𝑥 + 1
𝑥 − 1
. 
 
𝑃𝑒𝑙𝑎 𝑑𝑒𝑓𝑖𝑛𝑖çã𝑜 𝑑𝑒 𝑓𝑢𝑛çã𝑜 𝑑𝑒𝑟𝑖𝑣𝑎𝑑𝑎, 𝑡𝑒𝑚𝑜𝑠: