Question

Lim {(x+1)^5-1}/x
X➡0

Answer 1

Answer:

${\large{\overbrace{\underbrace{\purple{your \: answer:5 }}}}} \\ \\ ♡ dear *----*$

Step-by-step explanation:

$\bf\ lim \frac{ ({x + 1)}^{5} }{x} \\ x \mapsto0 \\ \\ \bf\ let \\ \bf\underline{\: x + 1 = y} \\ \\ \bf\red{so \: x \mapsto0} \\ \bf\pink{ y\mapsto1} \\ \\ \bf\ \implies \: lim \: \: \: \frac{ {y}^{5} - 1}{y - 1} \\ \: \: \: \: \: \: \: \bf\ \: y \mapsto1 \\ \\ \: we \: know \: that\\ \bf\red{lim \frac{ {x}^{n} - {a}^{n} }{x - a} = {na}^{n - 1}} \\ \bf\ x \mapsto \: a \\ \\ \bf\ \implies \: lim \frac{ {y}^{5} - {1}^{5} }{y - 1} \\ \: \: \: \: \: \: \: \: \: \: \bf\ \: x \mapsto 1 \\ \\\bf\ \implies5 \times {1}^{5 - 1} \\ \\\bf\ \implies5 \times {1}^{4} \\ \\ \bf\boxed{ \implies5 \times 1 =5 } \\ \\ \large\boxed{ \fcolorbox{red}{pink}{hope \: it \: help \: you}}$

Answer 2

Solution :-

$\lim _{x\to\:0} \frac{( {x + 1})^{5} - 1 }{x} \\ \\ putting \: x = 0 \\ \\ = \: \frac{( {0 + 1})^{5} - 1 }{0} \\ \\ = \frac{ {1}^{5} - 1 }{0} \\ \\ = \frac{1 - 1}{0} \\ \\ = \frac{0}{0} \\ \\ Since ,\: it \: is \: in \: the \: form \: of \: \frac{0}{0}$

Formula Used :-

$\lim _{x\to\:a} \frac{ {x}^{n} - {a}^{n} }{x - a} = n {a}^{n - 1}$

So,

$\lim _{x\to\:0} \frac{( {x + 5})^{5} - 1}{ x} \\ \\ \: let \: y = x + 1 \\ x = y - 1 \\ \\ here \: x \rightarrow 0 \\ then \: y \rightarrow0 + 1 \\ y \rightarrow1 \\ \\ Therefore, \: \\ \lim _{x\to\:0} \frac{( {x + 1})^{5} - 1}{x} = \lim _{y\to\:1} \frac{ {y}^{5} - 1}{y - 1} \\ \\ = \lim _{y\to\:1} \frac{ {y}^{5} - {1}^{5} }{y - 1} \\ \\ = 5 \times {1}^{5 - 1} \\ \\ = 5 \times {1}^{4} \\ \\ = 5 \\ \\ {\purple{\boxed{\large{\bold{\lim_{x\to\:0} \frac{ ({x + 1})^{5} - 1}{x} = 5 }}}}}$