CBSE Class 12 Sample Paper for 2021 Boards

Class 12
Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

## Find the value(s) of k so that the following function is continuous at π₯ = 0 f (x) = { 1- cos β‘kx / x sinβ‘xΒ  if xβ 0 1/2 if x=0

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### Transcript

Question 21 Find the value(s) of k so that following function is continuous at π₯ = 0, f (x) = {β((1 β cosβ‘ππ₯)/(π₯ sinβ‘π₯ ) ππ π₯β 0@ 1/2 ππ π₯=0)β€ Given that function is continuous at x = 0 π(π₯) is continuous at x = 0 i.e. limβ¬(xβ0) π(π₯)=π(0) Limit at x β 0 (πππ)β¬(π₯β0) f(x) = (πππ)β¬(ββ0) f(h) = limβ¬(hβ0) (1 β cosβ‘πβ)/(β (sinβ‘β) ) = limβ¬(hβ0) (2 sin^2β‘γπβ/2γ)/(β (sinβ‘β)) = limβ¬(hβ0) (2 sin^2β‘γπβ/2γ)/1 Γ1/(β (sinβ‘β)) = limβ¬(hβ0) (2 sin^2β‘γπβ/2γ)/(πβ/2)^2 Γ (πβ/2)^2/(β (sinβ‘β)) = limβ¬(hβ0) (2 sin^2β‘γπβ/2γ)/(πβ/2)^2 Γ (π^2 β^2)/(4β (sinβ‘β)) = limβ¬(hβ0) (2 sin^2β‘γπβ/2γ)/(πβ/2)^2 Γ (π^2 β)/(4 (sinβ‘β)) = π^2/2 limβ¬(hβ0) sin^2β‘γπβ/2γ/(πβ/2)^2 Γ β/sinβ‘β = π^2/2 limβ¬(hβ0) sin^2β‘γπβ/2γ/(πβ/2)^2 Γlimβ¬(hβ0) β/sinβ‘β = π^2/2 Γ 1 Γ 1 = π^π/π Now, limβ¬(xβ0) π(π₯)=π(0) π^2/2 = 1/2 π^2 =1 π =Β±π Hence, k = 1, β1