412 FROM l•:LEMl•:^■TAR^" aL(;ri',ra to ttik calculus. 



At tliis stage it will be well to prove the two very simple 

 propositions on limits — that the limit of a sum (jr product is the 

 ^uni or product of the limits. 



Let A he a. varying- (|uantity which by any process tends to 

 a limit a. Then at any >tage A-—a + a, where a vanishes in 

 the limit. So B = b +' jS. 



Therefore A+B-^a + b + a + /3 -. and AB^ ab + al3+ba + a3. 



Now in the limit fi + fS, af3, ha and a/3 vanish (assuming a, b linite). 

 Therefore Lt.(A+B) a \-h and Lt. (AB) ub. 



II. Rate of increase of sin.v 



is Lt.sin.v'— sinA- j. sinA-(A-'— a')/2 x' + x 



' -, ="-Lt. -—-, r^ cos -- 



X - X X -A (a ~x)l2 2 



J. sin(A-' — A-)/2 T^ -t'+a- ,. 1 ^ /^, , 



— i^t. -7^ -y — , Lt. cos ~, which— I. cosA (the angles 



(a —x):2 2 



being in circular measure). 



c ,u * r • L- . ■ T . tauA — tauA J. sin(A'— a) 

 bo, the rate 01 mcrease ot tauAisLt. -, "^Lt. 



a'--a (a' — a)cosa COSA' 



which =, as before, secV. 



1 1 T T- -1 r sni A -^ SUl A 



111. For sm A, Lt. : : — — , ^^ay. Lt 



A —A ' " sin^ — sin^ cos^ 



I 



VI-A^ 



Similarly tan 'a gives . 



I+A- 



IV. a' needs special treatment. 



D{h+v)=D}i + Dv without dithculty. 

 D{uv)==iiDv + vDii as usual. 



and D(//0=Lt.-^V=>^ Lt. - ^"~-^" . ^' 



A - A „ - „ .V - A 



=Lt.-^V==^'. Lt.'-^;^=/0.). Du. 



N — II X — A 



DK) =Lt.^^^=« l.t.'^'zi 

 ^/^^ x'-x 



The question now arises whether — ^ has a ' limit.' Our 

 previous work has given us no great acquaintance with such 

 an expression. It might tend to become or «; or ^-^ 



X 



might tend to one limit as x approaches by such steps as 

 i( /2.' I2-! V2O) ^"d to a different limit when t approaches it 

 ^y ±{h 3. 4- • ••)• If ^^■*" could ignore this latter difficulty, and 



