Tính giới hạn của các hàm số sau khi x→+∞ và khi x→−∞
a) f(x)=√x2−3xx+2 ;
b) f(x)=x+√x2−x+1 ;
c) f(x)=√x2−x−√x2+1 .
a) Khi x→+∞
limx→+∞√x2−3xx+2=limx→+∞|x|√1−3xx+2=limx→+∞x√1−3xx+2=limx→+∞√1−3x1+2x=1
Khi x→−∞
limx→−∞√x2−3xx+2=limx→−∞|x|√1−3xx+2=limx→−∞−x√1−3xx+2=limx→−∞−√1−3x1+2x=−1 ;
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b) Khi x→+∞
limx→+∞(x+√x2−x+1)=limx→+∞(x+x√1−1x+1x2)=limx→+∞x(1+√1−1x+1x2)=+∞
Khi x→−∞
limx→−∞(x+√x2−x+1)=limx→−∞x2−(x2−1+1)x−√x2−x+1=limx→−∞x−1x−√x2−x+1=limx→−∞x−1x−|x|√1−1x+1x2=limx→−∞x−1x+x√1−1x+1x2=limx→−∞1−1x1+√1−1x+1x2=12
c) Khi x→+∞
limx→+∞(√x2−x−√x2+1)=limx→+∞(x2−x)−(x2+1)√x2−x+√x2+1=limx→+∞−x−1x√1−1x+x√1+1x2=limx→+∞−1−1x√1−1x+√1+1x2=−12;
Khi x→−∞
limx→−∞(√x2−x−√x2+1)=limx→−∞(x2−x)−(x2+1)√x2−x+√x2+1=limx→−∞−x−1−x√1−1x−x√1+1x2=limx→−∞−1−1x−√1−1x−√1+1x2=12