Question

Let f(x) = `{{:(5^(1/x), x < 0),(lambda[x], x ≥ 0):}` and λ ∈ R, then at x = 0

Options

• f is discontinuous

• f is continuous only, if λ = 0

• f is continuous only, whatever λ may be

• None of these

Answer 1

f is discontinuous

Explanation:

As we know, A function f (x) is said to be continuous at a point x = a if `lim_(x -> a) f(x) = f(a)`, otherwise not continuous.

Thus f(x) is continuous at x = a iff `lim_(x -> a^-) f(x) = lim_(x -> a^+) f(x) = f(a)`

Since, `f(x) = {{:(5^(1/x)",", x < 0),(lambda[x]",", x ≥ 0):}` and λ ∈ R

RHL at x = 0: `lim_(x -> 0^+) f(x) = lim_(x -> 0^+) lambda[x] = lim_(h -> 0) lambda [h]` = 0

LHL at x = 0: `lim_(x -> 0^-) f(x) = lim_(x -> 0^-) 5^(1/x)`

= `lim_(h -> 0) 5^(-1/h) = 5^(oo) = oo`

And `f(0) = lambda[0]` = 0.

Since, LHL ≠ RHL

∴ f(x) is not continuous.