Question

Evaluate :

$\large \sf \lim_{x\to\ 0} \bigg( \frac{ {(x + 1) }^{5} - 1}{x} \bigg)$

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Answer 1

Question :-

$\sf \lim_{x\to\ 0} \bigg( \frac{ {(x + 1) }^{5} - 1}{x} \bigg) \:$

Answer :-

$\boxed{ \red{\sf \lim_{x\to\ 0} \bigg( \frac{ {(x + 1) }^{5} - 1}{x} \bigg)} = 5}$

Solution :-

$\sf \lim_{x\to\ 0} \bigg( \frac{ {(x + 1) }^{5} - 1}{x} \bigg) \: \\ \\ \sf{let \: t = x + 1} \\ \\ \rightarrow \: \sf \lim_{t\to\ 1} \bigg( \frac{ {t}^{5} - {1}^{5} }{t - 1} \bigg) \: \\ \\ \because \: \sf \lim_{x\to\ 0} \bigg( \frac{ {x }^{n} - {a}^{n} }{x - a} \bigg) \: = n {x}^{n - 1} \\ \\ \therefore \: \: a = 1 \: x = t \: n = 5 \\ \\ \rightarrow \: 5 \times {1}^{5 - 1} \\ \\ \rightarrow \: 5 \times 1 = 5$