Question 21 - CBSE Class 12 Sample Paper for 2021 Boards
Last updated at Oct. 26, 2020 by Teachoo

Find the value(s) of k so that the following function is continuous at π₯ = 0
f (x) = {
^{
1- cos β‘kx / x sinβ‘x if x≠0
}
_{
1/2 if x=0
}

_{
}

_{
}

Transcript

Question 21 Find the value(s) of k so that following function is continuous at π₯ = 0, f (x) = {β((1 β cosβ‘ππ₯)/(π₯ sinβ‘π₯ ) ππ π₯β 0@ 1/2 ππ π₯=0)β€
Given that function is continuous at x = 0
π(π₯) is continuous at x = 0
i.e. limβ¬(xβ0) π(π₯)=π(0)
Limit at x β 0
(πππ)β¬(π₯β0) f(x) = (πππ)β¬(ββ0) f(h)
= limβ¬(hβ0) (1 β cosβ‘πβ)/(β (sinβ‘β) )
= limβ¬(hβ0) (2 sin^2β‘γπβ/2γ)/(β (sinβ‘β))
= limβ¬(hβ0) (2 sin^2β‘γπβ/2γ)/1 Γ1/(β (sinβ‘β))
= limβ¬(hβ0) (2 sin^2β‘γπβ/2γ)/(πβ/2)^2 Γ (πβ/2)^2/(β (sinβ‘β))
= limβ¬(hβ0) (2 sin^2β‘γπβ/2γ)/(πβ/2)^2 Γ (π^2 β^2)/(4β (sinβ‘β))
= limβ¬(hβ0) (2 sin^2β‘γπβ/2γ)/(πβ/2)^2 Γ (π^2 β)/(4 (sinβ‘β))
= π^2/2 limβ¬(hβ0) sin^2β‘γπβ/2γ/(πβ/2)^2 Γ β/sinβ‘β
= π^2/2 limβ¬(hβ0) sin^2β‘γπβ/2γ/(πβ/2)^2 Γlimβ¬(hβ0) β/sinβ‘β
= π^2/2 Γ 1 Γ 1
= π^π/π
Now,
limβ¬(xβ0) π(π₯)=π(0)
π^2/2 = 1/2
π^2 =1
π =Β±π
Hence, k = 1, β1

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