Applicatian 1) \( \lim _{x \rightarrow 0^{+}} 3 x+7+\frac{1}{\sqrt{x}} \) 2) \( \lim _{x \rightarrow-\infty}\left(-3+\frac{1}{x}\right)\left(1+\frac{2}{x}\right) \) 3) \( \underset{\left.x \rightarrow-\infty \frac{1+\frac{4}{x^{3}}}{2-x^{5}} \quad 4\right) \frac{\ln }{x \rightarrow+\infty} \frac{\sqrt{x^{2}+5 x-3}}{x}}{x} \) \( \lim _{x \rightarrow+\infty} \sqrt{4 x^{2}-x+3}-2 x+1 \) 6) \( \lim _{x \rightarrow-\infty} \sqrt{x^{2}+x}+x \)

Evaluate the limit by following steps: - step0: Evaluate the limit: \(\lim _{x\rightarrow 0^{+}}\left(3x+7+\frac{1}{\sqrt{x}}\right)\) - step1: Rewrite the expression: \(\lim _{x\rightarrow 0^{+}}\left(3x\right)+\lim _{x\rightarrow 0^{+}}\left(7\right)+\lim _{x\rightarrow 0^{+}}\left(\frac{1}{\sqrt{x}}\right)\) - step2: Calculate: \(0+7+\left(+\infty\right)\) - step3: Calculate: \(7+\left(+\infty\right)\) - step4: Calculate: \(+\infty\) Calculate the limit \( \lim_{x \rightarrow -\infty} \frac{1 + \frac{4}{x^{3}}}{2 - x^{5}} \). Evaluate the limit by following steps: - step0: Evaluate the limit: \(\lim _{x\rightarrow -\infty}\left(\frac{1+\frac{4}{x^{3}}}{2-x^{5}}\right)\) - step1: Rewrite the expression: \(\frac{\lim _{x\rightarrow -\infty}\left(1+\frac{4}{x^{3}}\right)}{\lim _{x\rightarrow -\infty}\left(2-x^{5}\right)}\) - step2: Calculate: \(\frac{1}{\lim _{x\rightarrow -\infty}\left(2-x^{5}\right)}\) - step3: Calculate: \(\frac{1}{+\infty}\) - step4: Calculate: \(0\) Calculate the limit \( \lim_{x \rightarrow +\infty} \frac{\ln x}{x} \). Evaluate the limit by following steps: - step0: Evaluate using L'Hopital's rule: \(\lim _{x\rightarrow +\infty}\left(\frac{\ln{\left(x\right)}}{x}\right)\) - step1: Use the L'Hopital's rule: \(\lim _{x\rightarrow +\infty}\left(\frac{\frac{d}{dx}\left(\ln{\left(x\right)}\right)}{\frac{d}{dx}\left(x\right)}\right)\) - step2: Find the derivative: \(\lim _{x\rightarrow +\infty}\left(\frac{\frac{1}{x}}{1}\right)\) - step3: Simplify the expression: \(\lim _{x\rightarrow +\infty}\left(\frac{1}{x}\right)\) - step4: Rewrite the expression: \(\frac{1}{\lim _{x\rightarrow +\infty}\left(x\right)}\) - step5: Calculate: \(\frac{1}{+\infty}\) - step6: Divide the terms: \(0\) Calculate the limit \( \lim_{x \rightarrow -\infty} \left(\sqrt{x^{2} + x} + x\right) \). Evaluate the limit by following steps: - step0: Evaluate using transformations: \(\lim _{x\rightarrow -\infty}\left(\sqrt{x^{2}+x}+x\right)\) - step1: Multiply by the Conjugate: \(\lim _{x\rightarrow -\infty}\left(\frac{x}{\sqrt{x^{2}+x}-x}\right)\) - step2: Rearrange the terms: \(\lim _{x\rightarrow -\infty}\left(\frac{1\times x}{\left(-\sqrt{1+\frac{1}{x}}-1\right)x}\right)\) - step3: Reduce the fraction: \(\lim _{x\rightarrow -\infty}\left(\frac{1}{-\sqrt{1+\frac{1}{x}}-1}\right)\) - step4: Rewrite the expression: \(\frac{\lim _{x\rightarrow -\infty}\left(1\right)}{\lim _{x\rightarrow -\infty}\left(-\sqrt{1+\frac{1}{x}}-1\right)}\) - step5: Evaluate: \(\frac{1}{\lim _{x\rightarrow -\infty}\left(-\sqrt{1+\frac{1}{x}}-1\right)}\) - step6: Evaluate: \(\frac{1}{-2}\) - step7: Rewrite the fraction: \(-\frac{1}{2}\) Calculate the limit \( \lim_{x \rightarrow -\infty} \left(-3 + \frac{1}{x}\right)\left(1 + \frac{2}{x}\right) \). Evaluate the limit by following steps: - step0: Evaluate the limit: \(\lim _{x\rightarrow -\infty}\left(\left(-3+\frac{1}{x}\right)\left(1+\frac{2}{x}\right)\right)\) - step1: Rewrite the expression: \(\lim _{x\rightarrow -\infty}\left(-3+\frac{1}{x}\right)\times \lim _{x\rightarrow -\infty}\left(1+\frac{2}{x}\right)\) - step2: Calculate: \(-3\times 1\) - step3: Multiply: \(-3\) Calculate the limit \( \lim_{x \rightarrow +\infty} \left(\sqrt{4x^{2} - x + 3} - 2x + 1\right) \). Evaluate the limit by following steps: - step0: Evaluate using transformations: \(\lim _{x\rightarrow +\infty}\left(\sqrt{4x^{2}-x+3}-2x+1\right)\) - step1: Multiply by the Conjugate: \(\lim _{x\rightarrow +\infty}\left(\frac{3x+2}{\sqrt{4x^{2}-x+3}+2x-1}\right)\) - step2: Rearrange the terms: \(\lim _{x\rightarrow +\infty}\left(\frac{\left(3+\frac{2}{x}\right)x}{\left(\sqrt{4-\frac{1}{x}+\frac{3}{x^{2}}}+2-\frac{1}{x}\right)x}\right)\) - step3: Reduce the fraction: \(\lim _{x\rightarrow +\infty}\left(\frac{3+\frac{2}{x}}{\sqrt{4-\frac{1}{x}+\frac{3}{x^{2}}}+2-\frac{1}{x}}\right)\) - step4: Rewrite the expression: \(\frac{\lim _{x\rightarrow +\infty}\left(3+\frac{2}{x}\right)}{\lim _{x\rightarrow +\infty}\left(\sqrt{4-\frac{1}{x}+\frac{3}{x^{2}}}+2-\frac{1}{x}\right)}\) - step5: Evaluate: \(\frac{3}{\lim _{x\rightarrow +\infty}\left(\sqrt{4-\frac{1}{x}+\frac{3}{x^{2}}}+2-\frac{1}{x}\right)}\) - step6: Evaluate: \(\frac{3}{4}\) Voici les résultats des limites demandées : 1) \( \lim _{x \rightarrow 0^{+}} \left(3x + 7 + \frac{1}{\sqrt{x}}\right) = +\infty \) 2) \( \lim _{x \rightarrow -\infty} \left(-3 + \frac{1}{x}\right)\left(1 + \frac{2}{x}\right) = -3 \) 3) \( \lim _{x \rightarrow -\infty} \frac{1 + \frac{4}{x^{3}}}{2 - x^{5}} = 0 \) 4) \( \lim _{x \rightarrow +\infty} \frac{\ln x}{x} = 0 \) 5) \( \lim _{x \rightarrow +\infty} \left(\sqrt{4x^{2} - x + 3} - 2x + 1\right) = \frac{3}{4} \) 6) \( \lim _{x \rightarrow -\infty} \left(\sqrt{x^{2} + x} + x\right) = -\frac{1}{2} \) Si vous avez besoin d'explications supplémentaires sur l'un de ces résultats, n'hésitez pas à demander !