Given:" "f(x)=0.72^x-2 f(x)=0.72x−2
There are three prime conditions for this expression
Condition 1: x>0x>0
Condition 2: x<0x<0
Condition 3: x=0x=0
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color(blue)("Consider condition 3 ")x=0Consider condition 3 x=0
When x=0" "f(x)x=0 f(x) is defined as it has a definite value.
Thus color(blue)(x=0 " is NOT an asymptote")x=0 is NOT an asymptote
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color(blue)("Consider condition 2 ") x<0Consider condition 2 x<0
The f(x)f(x) takes the form 1/(0.72^x)-210.72x−2
When x=0": "f(x)=1-2" "x=0: f(x)=1−2 thus defined
As xx becomes bigger 0.72^20.722 becomes smaller so
1/(0.72^2)" "10.722 becomes bigger
Thus lim_(xtooo) f(x)->oo (infinity is not a number!)
color(brown)("Is this an asymptote? I have invited another mathematician to look at this bit!")
Gut feeling is that this is NOT an asymptote!!!!!
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color(blue)("Consider condition 1 ") x>0
As x becomes increasingly larger 0.72^x becomes increasingly smaller and smaller.
So lim_(xtooo) 0.72^x-2 -> -2color
color(blue)(-2" Is an asymptote")
