Question

Let f(x) = {((sinx + cosx)^cosecx, : - 1/2 ≤ x < 0), (a, : x = 0), ((e^1/x + e^2/x + e^3/|x|)/(ae^2/x + be^3/|x|), : 0 < x ≤ 1/2) If f(x) is continuous at x = 0, find the value of {e(a + b) + 2}.

Answer 1

We have lim_(x→0)  (sinx + cosx)^{cosec x} 

= lim _(x→0^–)  (1 + (sinx + cosx – 1))^{cosec x}

= e^{lim _(x→0–₎ (sinx + cosx – 1)/sinx}

Since f(x) is continuous at x = 0, so

lim _(x→0⁺)  f(x)  = lim _(x→0^–) f(x) = f(0)

1/b = e = a

Thus, a = e, b = 1/e

Hence, the value of {e(a + b) + 2}

= {e( e + 1/e ) + 2}

= (e² + 3)