10. \( \lim _{x \rightarrow 0} \frac{\sin 7 x}{x} \) 11. \( \lim _{x \rightarrow 0} \frac{\tan 3 x}{\sin 2 x} \)

Calculate the limit \( \lim_{x\rightarrow 0} \frac{\sin(7x)}{x} \). Evaluate the limit by following steps: - step0: Evaluate using L'Hopital's rule: \(\lim _{x\rightarrow 0}\left(\frac{\sin\left(7x\right)}{x}\right)\) - step1: Use the L'Hopital's rule: \(\lim _{x\rightarrow 0}\left(\frac{\frac{d}{dx}\left(\sin\left(7x\right)\right)}{\frac{d}{dx}\left(x\right)}\right)\) - step2: Find the derivative: \(\lim _{x\rightarrow 0}\left(\frac{7\cos\left(7x\right)}{1}\right)\) - step3: Simplify the expression: \(\lim _{x\rightarrow 0}\left(7\cos\left(7x\right)\right)\) - step4: Rewrite the expression: \(7\times \lim _{x\rightarrow 0}\left(\cos\left(7x\right)\right)\) - step5: Calculate: \(7\times 1\) - step6: Multiply: \(7\) Calculate the limit \( \lim_{x\rightarrow 0} \frac{\tan(3x)}{\sin(2x)} \). Evaluate the limit by following steps: - step0: Evaluate using L'Hopital's rule: \(\lim _{x\rightarrow 0}\left(\frac{\tan\left(3x\right)}{\sin\left(2x\right)}\right)\) - step1: Use the L'Hopital's rule: \(\lim _{x\rightarrow 0}\left(\frac{\frac{d}{dx}\left(\tan\left(3x\right)\right)}{\frac{d}{dx}\left(\sin\left(2x\right)\right)}\right)\) - step2: Find the derivative: \(\lim _{x\rightarrow 0}\left(\frac{3\sec^{2}\left(3x\right)}{2\cos\left(2x\right)}\right)\) - step3: Rewrite the expression: \(\frac{\lim _{x\rightarrow 0}\left(3\sec^{2}\left(3x\right)\right)}{\lim _{x\rightarrow 0}\left(2\cos\left(2x\right)\right)}\) - step4: Calculate: \(\frac{3}{\lim _{x\rightarrow 0}\left(2\cos\left(2x\right)\right)}\) - step5: Calculate: \(\frac{3}{2}\) La limite de \( \frac{\sin(7x)}{x} \) lorsque \( x \) tend vers 0 est égale à 7. La limite de \( \frac{\tan(3x)}{\sin(2x)} \) lorsque \( x \) tend vers 0 est égale à 1.5.