Question

Let f(x) = `{{:(3^(1/x);   x < 0","                "then at"  x = 0),(lambda[x];   x ≥ 0","   lambda ∈ "R"):}`

Options

• f is continuous whatever λ maybe

• f is discontinuous

• f is continuous only if λ = 0

• none of these

Answer 1

f is continuous whatever λ maybe

Explanation:

`lim_{x→0^-} f(x) = lim_{x→0^-} 3^{1/x} = lim_{h→0}3^{-1/h} = 0`

`lim_{x→0^+} f(x) = lim_{x→0^+}lambda[x] = 0,` for all λ ∈ R

f(0) = λ(0) = 0

∴ f is continuous at x = 0, whatever λ may be.