Problem 1
f(x)=log(1+∣x∣1−∣x∣)
1+∣x∣1−∣x∣>0=>1−∣x∣>0=>∣x∣<1 Domain:(−1,1)
x− intercept: y=0=>0=log(1+∣x∣1−∣x∣)=>1+∣x∣1−∣x∣=1
=>1−∣x∣=1+∣x∣=>x=0Point (0,0).
y− intercept: x=0=>y(0)=log(1+∣0∣1−∣0∣)=0
Point (0,0).
The graph passes through the origin.
1+∣x∣1−∣x∣=1,x=0
0<1+∣x∣1−∣x∣<1,x∈(−1,0)∪(0,1) Then f(x)<0,x∈(−1,0)∪(0,1) and f(0)=0.
Range:(−∞,∞)
x→−1+limf(x)=x→−1+limlog(1+∣x∣1−∣x∣)=−∞
x→1−limf(x)=x→1−limlog(1+∣x∣1−∣x∣)=−∞
Problem 2
x→∞lim(x2−3x−x2−5x+1)
=x→∞lim(x2−3x+x2−5x+1x2−3x−(x2−5x+1))
=x→∞lim(x2x2−x23x+x2x2−x25x+x21x2x−x1)
=x→∞lim(1−x3+1−x5+x212−x1)
=1−0+1−0+02−0=1
Problem 3
x→−∞lim1−2xx6−x2
=x→−∞lim1−2x−xx4−1
=x→−∞limx1−x2x−x4−1
=x→−∞lim2−x1x4−1=∞
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