Question

limx-5, x^k -5^k/x-5 = 500​

Answer 1

$\large\underline\blue{\bold{Given \:  Question :-  }}$

$\bf \:\lim_{x\to5}\dfrac{ {x}^{k} - {5}^{k} }{x - 5} = 500 \: then \: find \: k.$

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$\bf \:\large \red{AηsωeR } ✍$

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$\large\underline\blue{\bold{Formula \: Used :-  }}$

$\bf \:\lim_{x\to \: a}\dfrac{ {x}^{n} - {a}^{n} }{x - a} = n {a}^{n - 1}$

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$\large\underline\purple{\bold{Solution :- }}$

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$\bf \:\lim_{x\to5}\dfrac{ {x}^{k} - {5}^{k} }{x - 5} = 500$

$\sf\implies \:k {(5) }^{k - 1} = 500$

$\sf\implies \:k {(5) }^{k - 1} = 5 \times 5 \times 5 \times 4$

$\sf\implies \:k {(5) }^{k - 1} = 4 \times {5}^{3}$

$\sf\implies \:k {(5) }^{k - 1} = 4 \times {5}^{4 - 1}$

★ So, on comparing we get

$\bf\implies \:k = 4$

$\large{\boxed{\boxed{\sf{Hence,\sf \:\lim_{x\to5}\dfrac{ {x}^{k} - {5}^{k} }{x - 5} = 500 \implies k = 4}}}}$