Expert answers to "Find the limit limu→1u4−1u3−1"

shadsiei

Answered question

2020-12-16

Find the limit

lim_{u \to 1} \frac{u^{4} - 1}{u^{3} - 1}

Dora

Skilled2020-12-17Added 98 answers

Given,

lim_{u \to 1} \frac{u^{4} - 1}{u^{3} - 1}

On simplification, we get

lim_{u \to 1} \frac{u^{4} - 1}{u^{3} - 1} = lim_{u \to 1} \frac{(u^{2})^{2} - 1^{2}}{u^{3} - 1^{3}}

= lim_{u \to 1} \frac{(u^{2} - 1) (u^{2} + 1)}{(u - 1) (u^{2} + u + 1)}

[\begin{matrix} ∵ a^{2} - b^{2} = (a - b) (a + b) \\ a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2}) \end{matrix}]

= lim_{u \to 1} \frac{(u - 1) (u + 1) (u^{2} + 1)}{(u - 1) (u^{2} + u + 1)}

= lim_{u \to 1} \frac{(u + 1) (u^{2} + 1)}{(u^{2} + u + 1)}

= lim_{u \to 1} \frac{(1 + 1) (1^{2} + 1)}{(1^{2} + 1 + 1)}

= \frac{4}{3}

Jeffrey Jordon

Expert2022-08-24Added 2605 answers

Answer is given below (on video)

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