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Question
Find the values of p and q for which
f(x) = `{((1-sin^3x)/(3cos^2x),`
is continuous at x = π/2.
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Solution
f(x) = `{((1-sin^3x)/(3cos^2x),`
For continuity,
`lim_(x->(pi^-)/2)f(x)=lim_(x->(pi^+)/2)f(x)=f(pi/2)`
`lim_(x->(pi^-)/2)f(x)=lim_(x->(pi^-)/2)((1-sin^3x)/(3cos^2x))=lim_(x->(pi^-)/2)((1-sinx)(1+sin^2x+sinx))/(3[1-sin^2x])`
`lim_(x->pi/2)f(x)=lim_(x-pi/2)(1+sin^2x+sinx)/(3(1+sinx))=(1+1+1)/(3(2))=1/2`
Let `pi/2-x=theta=>x=pi/2-theta`
`lim_(x->pi^+)=lim_(theta->0)q[(1-sin(pi/2-theta))/(20)^2]=q/4lim_(theta->0)(1-costheta)/theta^2`
`=q/4lim_(theta->0)(2sin^2`
Now, `lim_(x->pi/2)f(x)=lim_(x->pi^+)f(x)=f(pi/2)`
`=>1/2=p=q/8`
`=>p=1/2 `
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