$\underset{x\to 0}{lim}\frac{\sqrt{x\+1}-1}{x}$ = $\frac{1}{2}$$$

Annotated stepwise solution via the Context Panel

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$\underset{x\to 0}{lim}\frac{\sqrt{x\+1}-1}{x}$$\stackrel{\text{show solution steps}}{\to }$$\begin{array}{lll}& & \text{Limit Steps}\\ & & \underset{x\to 0}{lim}\frac{\sqrt{x+1}-1}{x}\\ \text{▫}& & \text{1. Apply the}\mathbf{\text{L'Hôpital's Rule}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{L'Hôpital's Rule}}\text{rule}\\ & & \underset{x\to c}{lim}\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to c}{lim}\frac{\frac{\ⅆ}{\ⅆx}f\left(x\right)}{\frac{\ⅆ}{\ⅆx}g\left(x\right)}\\ & \text{◦}& \text{Rule applied}\\ & & \frac{\sqrt{x+1}-1}{x}=\frac{1}{2\sqrt{x+1}}\\ & & \text{This gives:}\\ & & \underset{x\to 0}{lim}\frac{1}{2\sqrt{x+1}}\\ \text{▫}& & \text{2. Apply the}\mathbf{\text{constant multiple}}\text{rule to the term}\underset{x\to 0}{lim}\frac{1}{2\sqrt{x+1}}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{constant multiple}}\text{rule}\\ & & \underset{x\to 0}{lim}Cf\left(x\right)=C\left(\underset{x\to 0}{lim}f\left(x\right)\right)\\ & \text{◦}& \text{This means:}\\ & & \underset{x\to 0}{lim}\frac{1}{2\sqrt{x+1}}=\frac{1}{2}\cdot \left(\underset{x\to 0}{lim}\frac{1}{\sqrt{x+1}}\right)\\ & & \text{We can rewrite the limit as:}\\ & & \frac{\left(\underset{x\to 0}{lim}\frac{1}{\sqrt{x+1}}\right)}{2}\\ \text{▫}& & \text{3. Apply the}\mathbf{\text{power}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{power}}\text{rule}\\ & & \underset{x\to a}{lim}{x}^n={\left(\underset{x\to a}{lim}x\right)}^n\\ & & \text{This gives:}\\ & & \frac{1}{2\sqrt{\underset{x\to 0}{lim}\left(x+1\right)}}\\ \text{▫}& & \text{4. Apply the}\mathbf{\text{sum}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{sum}}\text{rule}\\ & & \underset{x\to a}{lim}\left(f\left(x\right)+g\left(x\right)\right)=\left(\underset{x\to a}{lim}f\left(x\right)\right)+\left(\underset{x\to a}{lim}g\left(x\right)\right)\\ & & f\left(x\right)=x\\ & & g\left(x\right)=1\\ & & \text{This gives:}\\ & & \frac{1}{2\sqrt{\left(\underset{x\to 0}{lim}x\right)+\left(\underset{x\to 0}{lim}1\right)}}\\ \text{▫}& & \text{5. Apply the}\mathbf{\text{constant}}\text{rule to the term}\underset{x\to 0}{lim}1\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{constant}}\text{rule}\\ & & \mathrm{Limit}\left(C\,x\right)=C\\ & \text{◦}& \text{This means}\\ & & \underset{x\to 0}{lim}1=1\\ & & \text{We can now rewrite the limit as:}\\ & & \frac{1}{2\sqrt{\left(\underset{x\to 0}{lim}x\right)+1}}\\ \text{▫}& & \text{6. Apply the}\mathbf{\text{identity}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{identity}}\text{rule}\\ & & \underset{x\to a}{lim}x=a\\ & & \text{This gives:}\\ & & \frac{1}{2}\end{array}$