Determining the unknown constant? Let Determine a v ? alue of? (if possible) for ? which f ? ?(1) exists.
SOLUTION STEP 1 The given function is We have to determine the value of a such that f(1)exists. This implies that the left hand and the right hand derivatives are equal for f(1); ie, f+(1) = f (1) Therefore f(1+h)f(1) f(1+h)f(1) lim + h = lim h h0 h0 2 2 2 lim a(1+h)22(1= lim 2(1+h) 2(1) + h h h0 h0 a+ah4 2(1+h +2h)2 lim h = lim h h0+ h0 2 lim a+ah4= lim 2+2h +4h2 h0+ h h0 h a+ah4 2h +4h lim + h = lim h h0 h0 a+ah4 lim + h = lim h + 4 h0 h0 lim a+ah4= 4……………(1) h0+ h On the left hand side lim a+ah4 cannot be calculated as it is in the indeterminate + h h0 form.Thus here we use the L’hospitals rule. d a+ah4 dh(a+ah4) a Thus lim + h = lim + dh = lim +1 = a h0 h0 h0 Thus from (1) we get a=4.