Q. A straight line is said to be an asymptote to the curve y=f(x) if the ⊥ distance of point P(x, y) on the curve from the line tends to zero when x→∞ or y→∞ or both x & y→∞
(a) Asymptotes parallel to y-axis:
The line x=k is asymptote to the curve y=f(x) If
limx→k+f(x)=∞ or −∞ & limx→k−f(x)=+∞ or −∞
b) Asymptotes parallel to x-axis:
The line y=k is asymptotes to the curve y=f(x)
if limx→∞y=k or limx→−∞y=k
(c) Oblique asymptotes:
Let y=mx+c be an asymptote of the curve y=f(x), then from any point P(x, y) on the curve, the ⊥ distance is |y−mx−c|√1+m2 which →0 as x tends to +∞ or +∞
Thus y−mx−c→0 as x→+∞ or −∞
Again y−mx−c=x(yx−cx−m)=0
As x→∞ and y−mx−c→0
So limx→±∞(yx−cx−m)=0 ⇒ limx→±∞yx=m
and limx→±∞y−mx−c=0 ⇒ limx→±∞y−mx=c
So we come to the fact that if y=mx+c is an asymptote to the curve y=f(x). Then as x→+∞ or −∞
limx→∞yx=m=limx→∞f(x)x & limx→∞=y−mx=limx→∞{f(x)−mx}=c
On the basis of above information answer the following questions. The vertical & horizontal asymptotes to the curve
y=exx are respectively
