$\underset{x\to 0}{lim}\frac{{\sin }^2\left(x\right)}{x}$ = $0$$$

Annotated stepwise solution via the Context Panel

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$\underset{x\to 0}{lim}\frac{{\sin }^2\left(x\right)}{x}$$\stackrel{\text{show solution steps}}{\to }$$\begin{array}{lll}& & \text{Limit Steps}\\ & & \underset{x\to 0}{lim}\frac{{\sin \left(x\right)}^2}{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{{\sin \left(x\right)}^2}{x}=2\sin \left(x\right)\cos \left(x\right)\\ & & \text{This gives:}\\ & & \underset{x\to 0}{lim}2\sin \left(x\right)\cos \left(x\right)\\ \text{▫}& & \text{2. Apply the}\mathbf{\text{constant multiple}}\text{rule to the term}\underset{x\to 0}{lim}2\sin \left(x\right)\cos \left(x\right)\\ & \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}2\sin \left(x\right)\cos \left(x\right)=2\cdot \left(\underset{x\to 0}{lim}\sin \left(x\right)\cos \left(x\right)\right)\\ & & \text{We can rewrite the limit as:}\\ & & 2\left(\underset{x\to 0}{lim}\sin \left(x\right)\cos \left(x\right)\right)\\ \text{▫}& & \text{3. Apply the}\mathbf{\text{product}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{product}}\text{rule}\\ & & \underset{x\to a}{lim}f\left(x\right)g\left(x\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)=\sin \left(x\right)\\ & & g\left(x\right)=\cos \left(x\right)\\ & & \text{This gives:}\\ & & 2\left(\underset{x\to 0}{lim}\sin \left(x\right)\right)\left(\underset{x\to 0}{lim}\cos \left(x\right)\right)\\ \text{▫}& & \text{4. Evaluate the limit of}\sin \text{(}x\text{)}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{sin}\text{rule}\\ & & \underset{x\to a}{lim}\sin \left(f\left(x\right)\right)=\sin \left(\underset{x\to a}{lim}f\left(x\right)\right)\\ & & \text{This gives:}\\ & & 0\end{array}$

Table 3.9.2(a)   Annotated stepwise solution using the Limit Methods tutor