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Question
If `lim_(x -> 1) (x^4 - 1)/(x - 1) = lim_(x -> k) (x^3 - l^3)/(x^2 - k^2)`, then find the value of k.
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Solution
Given that `lim_(x -> 1) (x^4 - 1)/(x - 1) = lim_(x -> k) (x^3 - l^3)/(x^2 - k^2)`
⇒ `4(1)^(4 - 1) = lim_(x -> k) ((x - k)(x^2 + k^2 + kx))/((x - k)(x + k))`
⇒ 4 = `lim_(x -> k) (x^2 + k^2 + kx)/(x + k)`
⇒ 4 = `(k^2 + k^2 + k^2)/(2k)`
⇒ 4 = `(3k^2)/(2k)`
⇒ 4 = `3/2 k`
⇒ k = `8/3`
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