Question

please solve this problem...​prove the problem

Answer 1

Given :-

$\displaystyle \lim_{ \rm \: x \to \: 0}{ \frac{ \sqrt{x + 1} - 1}{ \rm \: x} } = \dfrac{1}{2}$

To prove :

L.H.S = R.H.S

Proof :

R.H.S = 1/2

$\rm L.H.S = \displaystyle \lim_{ \rm \: x \to \: 0}{ \frac{ \sqrt{x + 1} - 1}{ \rm \: x} }$

$\rm \implies\displaystyle \lim_{ \rm \: x \to \: 0}{ \dfrac{ \sqrt{x + 1} - 1}{ \rm \: x} } \\ \\ \\ \rm \implies\displaystyle \lim_{ \rm \: x \to \: 0}{ \dfrac{ \sqrt{x + 1} - 1}{ \rm \: x} } \times \dfrac{ \sqrt{x + 1} + 1}{ \rm \:\sqrt{x + 1} + 1 } \\ \\ \\ \rm \implies\displaystyle \lim_{ \rm \: x \to \: 0}{ \dfrac{ \rm{x + 1} - 1}{ \rm \: x \: (\rm \:\sqrt{x + 1} + 1 )}} \\ \\ \\ \rm \implies\displaystyle \lim_{ \rm \: x \to \: 0}{ \dfrac{ \rm{x }}{ \rm \: x \: (\rm \:\sqrt{x + 1} + 1 )}} \\ \\ \\ \rm \implies\displaystyle \lim_{ \rm \: x \to \: 0}{ \dfrac{ \rm{1}}{ \rm \: \: (\rm \:\sqrt{x + 1} + 1 )}} \\ \\ \\ \rm \implies\displaystyle { \dfrac{ \rm{1}}{ \rm \: \: (\rm \:\sqrt{0 + 1} + 1 )}} \\ \\ \\ \rm \implies\displaystyle { \dfrac{ \rm{1}}{ \rm \: \: (\rm \:1 + 1 )}} \\ \\ \\ \rm \implies\displaystyle { \dfrac{ \rm{1}}{ \rm2 }} \\ \\ \\ \therefore\displaystyle \lim_{ \rm \: x \to \: 0}{ \frac{ \sqrt{x + 1} - 1}{ \rm \: x} } = \dfrac{1}{2}$

L.H.S = R.H.S

hence proved

Answer 2

Step-by-step explanation:

$\large\bf{\underline{\underline{We\ have:-}}}$

$\bf : \implies {\displaystyle \lim_{\bf x \to \ 0}\ \bf \dfrac{\sqrt{x\ +\ 1}\ -\ 1}{x}\ =\ \dfrac{1}{2}}$

$\large\bf{\underline{\underline{To\ prove:-}}}$

$\bf : \implies {L.H.S\ =\ R.H.S}$

$\large\bf{\underline{\underline{Solution:-}}}$

$\bf : \implies {L.H.S\ =\ \displaystyle \lim_{\bf x \to \ 0}\ \bf \dfrac{\sqrt{x\ +\ 1}\ -\ 1}{x}} \\ \\ \\ \bf : \implies {\displaystyle \lim_{\bf x \to \ 0}\ \bf \dfrac{\sqrt{x\ +\ 1}\ -\ 1}{x}\ \times\ \dfrac{\sqrt{x\ +\ 1}\ +\ 1}{\sqrt{x\ +\ 1}\ +\ 1}} \\ \\ \\ \bf : \implies {\displaystyle \lim_{\bf x \to \ 0}\ \bf \dfrac{x\ +\ 1\ -\ 1}{x\ (\sqrt{x\ +\ 1}\ +\ 1)}} \\ \\ \\ \bf : \implies {\displaystyle \lim_{\bf x \to \ 0}\ \bf \dfrac{x}{x\ (\sqrt{x\ +\ 1}\ +\ 1)}} \\ \\ \\ \bf : \implies {\displaystyle \lim_{\bf x \to \ 0}\ \bf \dfrac{1}{x\ (\sqrt{x\ +\ 1}\ +\ 1)}} \\ \\ \\ \bf : \implies {\dfrac{1}{(\sqrt{0\ +\ 1}\ +\ 1)}} \\ \\ \\ \bf : \implies {\dfrac{1}{(1\ +\ 1)}} \\ \\ \\ \bf : \implies {\purple{\underline{\boxed{\bf R.H.S\ =\ \dfrac{1}{2}}}}}\bigstar$

⠀⠀⠀

$\begin{gathered}\qquad\qquad\boxed{\bf{\mid{\overline{\underline{\pink{\bigstar Hence\ Proved \bigstar}}}}}\mid}\\\\\end{gathered}$