Calculer \( \lim _{x \rightarrow 0} \frac{\sin 3 x}{\sin 5 x} ; \lim _{x \rightarrow 0} \frac{\tan x}{x} \)

Calculate or simplify the expression \( \lim_{x \rightarrow 0} \frac{\sin(3x)}{\sin(5x)} \). Evaluate the limit by following steps: - step0: Evaluate using L'Hopital's rule: \(\lim _{x\rightarrow 0}\left(\frac{\sin\left(3x\right)}{\sin\left(5x\right)}\right)\) - step1: Use the L'Hopital's rule: \(\lim _{x\rightarrow 0}\left(\frac{\frac{d}{dx}\left(\sin\left(3x\right)\right)}{\frac{d}{dx}\left(\sin\left(5x\right)\right)}\right)\) - step2: Find the derivative: \(\lim _{x\rightarrow 0}\left(\frac{3\cos\left(3x\right)}{5\cos\left(5x\right)}\right)\) - step3: Rewrite the expression: \(\frac{\lim _{x\rightarrow 0}\left(3\cos\left(3x\right)\right)}{\lim _{x\rightarrow 0}\left(5\cos\left(5x\right)\right)}\) - step4: Calculate: \(\frac{3}{\lim _{x\rightarrow 0}\left(5\cos\left(5x\right)\right)}\) - step5: Calculate: \(\frac{3}{5}\) Calculate or simplify the expression \( \lim_{x \rightarrow 0} \frac{\tan(x)}{x} \). Evaluate the limit by following steps: - step0: Evaluate using L'Hopital's rule: \(\lim _{x\rightarrow 0}\left(\frac{\tan\left(x\right)}{x}\right)\) - step1: Use the L'Hopital's rule: \(\lim _{x\rightarrow 0}\left(\frac{\frac{d}{dx}\left(\tan\left(x\right)\right)}{\frac{d}{dx}\left(x\right)}\right)\) - step2: Find the derivative: \(\lim _{x\rightarrow 0}\left(\frac{\sec^{2}\left(x\right)}{1}\right)\) - step3: Simplify the expression: \(\lim _{x\rightarrow 0}\left(\sec^{2}\left(x\right)\right)\) - step4: Rewrite the expression: \(\left(\lim _{x\rightarrow 0}\left(\sec\left(x\right)\right)\right)^{2}\) - step5: Calculate: \(1^{2}\) - step6: Evaluate the power: \(1\) La limite de \( \frac{\sin 3x}{\sin 5x} \) lorsque \( x \) tend vers 0 est égale à \( \frac{3}{5} \). La limite de \( \frac{\tan x}{x} \) lorsque \( x \) tend vers 0 est égale à 1.