Exercicios Calculo (186)

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202 
 
𝑸𝒖𝒆𝒔𝒕ã𝒐 𝟏. 
 
𝑎) 𝑆𝑎𝑏𝑒𝑛𝑑𝑜 𝑞𝑢𝑒 lim
𝑥→0
(𝑓(𝑥) + 𝑔(𝑥))
2
= 4 𝑒 lim
𝑥→0
(𝑓(𝑥) − 𝑔(𝑥))
2
= 2 𝑒𝑛𝑐𝑜𝑛𝑡𝑟𝑒 
lim
𝑥→0
(𝑓(𝑥)𝑔(𝑥)). 
 
𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑒 𝑜 𝑠𝑒𝑔𝑢𝑖𝑛𝑡𝑒 𝑞𝑢𝑜𝑐𝑖𝑒𝑛𝑡𝑒: 
 
(𝑓(𝑥) + 𝑔(𝑥))
2
(𝑓(𝑥) − 𝑔(𝑥))
2
=
[𝑓(𝑥)]2 + 2. 𝑓(𝑥).𝑔(𝑥) + [𝑔(𝑥)]2
[𝑓(𝑥)]2 − 2. 𝑓(𝑥).𝑔(𝑥) + [𝑔(𝑥)]2
= 1 +
4𝑓(𝑥)𝑔(𝑥)
(𝑓(𝑥) − 𝑔(𝑥))
2
 
 
𝐶𝑜𝑚𝑜 lim
𝑥→0
(𝑓(𝑥)+ 𝑔(𝑥))
2
 𝑒 lim
𝑥→0
(𝑓(𝑥) − 𝑔(𝑥))
2
 𝑒𝑥𝑖𝑠𝑡𝑒𝑚,𝑐𝑜𝑚 𝑒𝑠𝑡𝑒 ú𝑙𝑡𝑖𝑚𝑜 
𝑑𝑖𝑓𝑒𝑟𝑒𝑛𝑡𝑒 𝑑𝑒 𝑧𝑒𝑟𝑜,𝑒𝑛𝑡ã𝑜 𝑜 𝑙𝑖𝑚𝑖𝑡𝑒 𝑑𝑜 𝑞𝑢𝑜𝑐𝑖𝑒𝑛𝑡𝑒 é 𝑜 𝑞𝑢𝑜𝑐𝑖𝑒𝑛𝑡𝑒 𝑑𝑜𝑠 𝑙𝑖𝑚𝑖𝑡𝑒𝑠 𝑒, 
𝑐𝑜𝑚𝑜 lim
𝑥→0
1 𝑒𝑥𝑖𝑠𝑡𝑒, 𝑐𝑜𝑛𝑐𝑙𝑢𝑖-𝑠𝑒 𝑝𝑒𝑙𝑜 𝑠𝑒𝑔𝑢𝑛𝑑𝑜 𝑚𝑒𝑚𝑏𝑟𝑜 𝑑𝑎 𝑖𝑔𝑢𝑎𝑙𝑑𝑎𝑑𝑒 𝑎𝑐𝑖𝑚𝑎 
𝑞𝑢𝑒 lim
𝑥→0
(𝑓(𝑥)𝑔(𝑥)) 𝑡𝑎𝑚𝑏é𝑚 𝑒𝑥𝑖𝑠𝑡𝑒. 𝐿𝑜𝑔𝑜, 
 
lim
𝑥→0
(𝑓(𝑥) + 𝑔(𝑥))
2
(𝑓(𝑥) − 𝑔(𝑥))
2
= lim
𝑥→0
1 + lim
𝑥→0
4. 𝑓(𝑥)𝑔(𝑥)
(𝑓(𝑥) − 𝑔(𝑥))
2
 
 
lim
𝑥→0
(𝑓(𝑥) + 𝑔(𝑥))
2
lim
𝑥→0
(𝑓(𝑥) − 𝑔(𝑥))
2
= lim
𝑥→0
1 +
lim
𝑥→0
(4.𝑓(𝑥)𝑔(𝑥))
lim
𝑥→0
(𝑓(𝑥) − 𝑔(𝑥))
2
 
 
4
2
= 1+
4 lim
𝑥→0
(𝑓(𝑥)𝑔(𝑥))
2
 
 
2 = 1+ 2 lim
𝑥→0
(𝑓(𝑥)𝑔(𝑥)) 
 
lim
𝑥→0
(𝑓(𝑥)𝑔(𝑥)) =
1
2
 
 
𝑏) 𝐶𝑎𝑙𝑐𝑢𝑙𝑒 lim
𝑥→∞
(√𝑥 + √𝑥 + √𝑥 −√𝑥). 
 
lim
𝑥→∞
(√𝑥 + √𝑥 + √𝑥 −√𝑥) = lim
𝑥→∞
𝑥 + √𝑥 + √𝑥 − 𝑥
√𝑥 +√𝑥 +√𝑥 + √𝑥
 
 
 = lim
𝑥→∞
√𝑥 + √𝑥
√𝑥 +√𝑥 +√𝑥 + √𝑥