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Question
Select the correct answer from the given alternatives.
`lim_(x → π/3) ((tan^2x - 3)/(sec^3x - 8))` =
Options
1
`1/2`
`1/3`
`1/4`
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Solution
`lim_(x → π/3) ((tan^2x - 3)/(sec^3x - 8)) = 1/3`
Explanation:
`lim_(x → π/3) ((tan^2x - 3)/(sec^3x - 8))`
`= lim_(x → π/3) ((sec^2x - 1 - 3)/(sec^3x - 8)) ...[(tan^2 x + 1 = sec^2 x),(∵ tan^2 x = sec^2x - 1)]`
`= lim_(x → π/3) ((sec^2x - 4)/(sec^3x - 8))`
`= lim_(x → π/3) ((sec^2x - (2)^2)/(sec^3x - (2)^3))`
`= lim_(x → π/3) ((secx - 2)(secx + 2))/((sec x - 2)(sec^2 x + 2sec x + 4)) ...[(a^2 - b^2 = (a - b)(a + b)),(a^3 - b^3 = (a - b)(a^2 + ab + b^2))]`
`= lim_(x → π/3) (secx + 2)/(sec^2 x + 2sec x + 4)`
`= (sec π/3 + 2)/((sec π/3)^2 + 2sec π/3 + 4)`
`= (2 + 2)/((2)^2 + 2(2) + 4)`
`= 1/3`
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