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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62. Eliminate the indeterminate functions from the equation z = xm F () -.Y
63. From the equation z = F(y + ax)f(y-ax), it is
required to eliminate the indeterminate functions F and f.
64. Eliminate the indeterminate, exponential, &c. functions from the following equations:
Cy
(1) -Z=^Y +f: (2) z - +f(ax -by):
\x/ a
(3) z x 2 +f (y + log x): (4) z=ysin-1 + (y)
(5) z -=f (y)- y/ '2 (6) log z-f (y) -
(7) (-y) Z=e-yf(xy-y)': (8) sin =f (z + \/ -:
(9) z=fl (log +2 /y): (10) z=log-l{y- +- L
(11) z=/ 2 + y2/f {tan1 - log 2-2+ y:
y
(12) Log z=b (y-a ) +log {f(y+-ax)+ P(y-ax)}:
(13) = ( )log (x2 + y2) - - (+6xy2-6y3 tan' -
6y 9y y
+ ~ (2 + 3y2)f (X) + (y):
(14) p+aq=f Y() ((15) ap-bq=f(ay+bx).
(16) p = qf (: (17) P+ q=(p -q) f ():
(18) z=V/yf (+P (xy)' (19) z=e'{(y —ax)+0 y}.
65. From the equation z = cx +f (fv + y), it is required
to prove that r2 - t2 = rt - s2.
66. From the equation - y + ay =0, it is required to
find the values ofy2 and y2 by Lagrange's Theorem.