If limx→1x4-1x-1=limx→kx3-l3x2-k2, then find the value of k. - Mathematics

Question

If $lim_{x \to 1} \frac{x^{4} - 1}{x - 1} = lim_{x \to k} \frac{x^{3} - l^{3}}{x^{2} - k^{2}}$ , then find the value of k.

Sum

Solution

Given that $lim_{x \to 1} \frac{x^{4} - 1}{x - 1} = lim_{x \to k} \frac{x^{3} - l^{3}}{x^{2} - k^{2}}$

⇒ $4 {(1)}^{4 - 1} = lim_{x \to k} \frac{(x - k) (x^{2} + k^{2} + k x)}{(x - k) (x + k)}$

⇒ 4 = $lim_{x \to k} \frac{x^{2} + k^{2} + k x}{x + k}$

⇒ 4 = $\frac{k^{2} + k^{2} + k^{2}}{2 k}$

⇒ 4 = $\frac{3 k^{2}}{2 k}$

⇒ 4 = $\frac{3}{2} k$

⇒ k = $\frac{8}{3}$

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Concept of Limits - Algebra of Limits

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Chapter 13: Limits and Derivatives - Exercise [Page 240]

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If limx→1x4-1x-1=limx→kx3-l3x2-k2, then find the value of k. - Mathematics

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