An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

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119. If ym - axm-l_ -xm = 0 be the equation of a curve;
a a
then will AW=- and AZ -: required a proof.
m m
120. In the curve whose equation is aym + f= bxm (a + x)"
it is required to prove that
n n
AW= -- a and AZ= (a+n-lb)T^+n.
m+-n m+ n
121. If the equation to a curve be x3 + y3 = a3; then will
A W= 0 and AZ =0, and the asymptote passes through the
origin, making 135~ with the axis of x: required a proof.
122. If (a - x) y2 = (a + x) x2 be the equation of a curve;
then will AW:= -a and AZ= + co, and the asymptote is
parallel to the axis of y: required a proof.
123. If (a2 - x2I") y2 = (a2 + V) be the equation of a
curve; then will AW= +a and AZ= + co, so that the
asymptotes are parallel to the axis of y: required a proof.
124. If the equation to a curve be (a+c+x) y = b (c+x);
then will AW= co and AZ=b when x is infinite, and the
asymptote is parallel to the axis of xB: also, A W= a+ c and
AZ =-co when y is infinite, and the asymptote is parallel to
the axis of y: required a proof.
125. If y2= a be the equation of a curve; then
iv-a
will AW= co and AZ= + 1, or there are two asymptotes
parallel to the axis of x and equi-distant from it: again,
A W= a and AZ = o, or there is another asymptote parallel
to the axis of y: required a proof.
126. Apply the principles of Article (151) to determine
the position of the rectilineal asymptote of the curve whose
equation is x5 + a3xy - y 0.
127. Find the equation to the rectilineal asymptote of the
hyperbola defined by the equation =
hyperbola defined by the equation - - = -1.
b' a'