Let f(x) = y = a*b^x -------------(i)

And g(x) = y = 1 +x

If f(x) and g(x) are tangent at x=0, then then their y's there are the same.

So,

f(0) = g(0)

a*b^0 = 1 +0

a*1 = 1

a = 1 -----------------answer.

So,

f(x) = y = 1*b^x = b^x -------(ii)

If f(x) and g(x) are tangent at x=0, then their slopes are equal at x=0.

slope, m = dy/dx

For g(x) = 1 +x,

m2 = g'(x) = 1 --------**

For f(x) = b^x,

m1 = f'(x) = ln(b) *b^x

At x=0,

m1 = m2

ln(b) *b^x = 1

ln(b) *b^0 = 1

ln(b) *1 = 1

ln(b) = 1

b = e^1

b = e ------------------------answer.