sin (a + y) = sina + x cos a + + n ... n-1 (− 1)"−1 (1 + n3) 2 cos {a-(n - 1) cot ̄1n} + &c. 6. Given a -y+x log y = 0, find sin y in powers of x. m ny 7. Given y z + xye", expand ye in powers of x. = 8. Given y=z+x sin y, expand sin y and sin 2y in powers of x. 9. Given y = log (z+x cos y), expand e' in powers of x. 10. From the equation xy* + 2xy3 + 3xy2+ 2y +1=0 determine y in ascending powers of x. π 4 +x sin log y 11. If y=e** find the first four terms of the expansion of cos log y in powers of x. In the same manner as we deduced the expansion of y” from the equation y = х 1 + √√(1 − x2) we may deduce the expansion of any other function of y; for example take logy. Thus ย logy = log z+x+... + x" 1 dr 2 [r 2r dzr-1 (22-1) + ...... where after the differentiations are performed we must put x I for z. Thus 2 The expansions which this example has furnished are of some importance in mathematics. 5. If x=ye", expand sin (a+y) in powers of x. We have given where coton, by a process similar to that in Art. 81. Putting z = 0 in this, we have for the required expansion sin (a + y) = sin a + x cos a + ... n-1 + (− 1)~~1 (1 + n3) cos (a− (n − 1) cot ̄1n} + &c. n 6. Given a -y+x log y = 0, find sin y in powers of x. m ny 7. Given y z + xye", expand ye" in = 8. Given y=z+x sin y, expand sin y and sin 2y in powers of x. 9. Given y = log (z+x cos y), expand e" in powers of x. 10. From the equation xy* + 2xy3 + 3xy2 + 2y +1=0 determine y in ascending powers of x. 12. If y3+my+nyx, shew that one value of y is CHAPTER X. LIMITING VALUES OF FUNCTIONS WHICH ASSUME AN 144. In the equation, limit of sin 0 = 1 when 0 diminishes indefinitely, we have an example of a fraction which approaches a finite limit when the numerator and denominator each tend to the limit zero. The object of this chapter is to find the limit of any fraction of which the numerator and denominator ultimately vanish, and also the limiting value of some other indeterminate forms. such a fraction that both numerator and denominator vanish when xa; it is required to find the limit towards which the above fraction tends as x approaches the limit a. We have proved in Art. 92 that $ (a + h) − $ (a) = hp' (a + Oh), ¥ (a + h) − ¥ (a) = hf' (a + 0 ̧h). If then (a) = 0 and (a) = 0, we have, by division, |