Question

lim xtend to 0 root 1+x-1divided x​

Answer 1

Answer:

0 × 1×1 +-1/

0×1+-1 /

0 + - 1 /

= -1

Answer 2

$\bf \: \displaystyle \rm \ \lim_{ x\to0} \dfrac{ \sqrt{1 + x} - 1}{ x}$

☆ On substituting the value of x directly, we get indeterminant form. So, on rationalizing, we get

$\bf \: = \displaystyle \rm \ \lim_{ x\to0} \bigg( \dfrac{ \sqrt{1 + x} - 1}{ x} \times \dfrac{ \sqrt{1 + x} + 1 }{ \sqrt{1 + x} + 1} \bigg)$

$\bf \: = \displaystyle \rm \ \lim_{ x\to0} \dfrac{1 + x - 1 }{ x( \sqrt{1 + x} + 1) }$

$\bf \: = \displaystyle \rm \ \lim_{ x\to0} \dfrac{x }{x( \sqrt{1 + x} + 1) }$

$\bf \: = \displaystyle \rm \ \lim_{ x\to0} \dfrac{ 1}{ \sqrt{1 + x} + 1 }$

$\bf \:  = \dfrac{1}{2}$

$\large{\boxed{\boxed{\bf{Hence \: \bf \: \displaystyle \rm \ \lim_{ x\to0} \dfrac{ \sqrt{1 + x} - 1}{ x} = \dfrac{1}{2} }}}}$