Let L = lim    [a - sqrt(a- x) - (x/4)] /  x, a > 0.

If L is finite, then

a). a = 2; L = 1/64
b). a = 1; L = 1/64
c). a = 3; L = 1/32
d). a = 1; L = 1/32

Dear Student,
Please find below the solution to the asked query:

L=limx0 a-a2-x2-x24x4=limx0 a-x24 -a2-x2x4=limx0 a-x24 -a2-x2x4a-x24 +a2-x2a-x24 +a2-x2=limx0 a-x24 2-a2-x2x41a-x24 +a2-x2=limx0 a2+x416-2ax24 -a2+x2x41a-x24 +a2-x2=limx0 x416-ax22 +x2x41a-0 +a2-0=limx0 x416+x21-a2x41a+aFor limit to be finite, x4 in denominator must get cancelled, hence coefficientof x2 in numerator must be 0.1-a2=0a=2HenceL=limx0 x416+0x412+2=limx0 11614=116×4=164Hence a=2 and L=164

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