Question

Find the value(s) of 'λ' if the function

f(x) = `{{:((sin^2 λx)/x^2",", if x ≠ 0  "is continuous at"  x = 0.),(1",", if x = 0):}`

Answer 1

f(x) = `{{:((sin^2 λx)/x^2",",  x ≠ 0),(1",",  x = 0):}`

For continuity at x = 0

`lim_(x rightarrow 0^-) f(x) = lim_(x rightarrow 0^+) f(x)` = f(x)

`lim_(x rightarrow 0^-) (sin^2 λx)/(λ^2x^2) xx λ^2 = lim_(x rightarrow 0^-) 1 xx λ^2` = λ²

`lim_(x rightarrow 0^+) (sin^2 λx)/(λ^2x^2) xx λ^2 = lim_(x rightarrow 0^+) λ^2` = λ²

f(0) = 1

Since f(x) is continuous.

λ² = 1

`\implies` λ = ± 1.