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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):}`
Solution
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.
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