c) \( \lim _{x \rightarrow+\infty} \frac{4 x+3}{\sqrt{2 x^{2}+3 x}} \)

Calculate the limit \( \lim_{x\rightarrow+\infty} \frac{4x+3}{\sqrt{2x^2+3x}} \). Evaluate the limit by following steps: - step0: Evaluate using transformations: \(\lim _{x\rightarrow +\infty}\left(\frac{4x+3}{\sqrt{2x^{2}+3x}}\right)\) - step1: Rewrite the expression: \(\lim _{x\rightarrow +\infty}\left(\frac{\left(4+\frac{3}{x}\right)x}{\sqrt{2+\frac{3}{x}}\times x}\right)\) - step2: Reduce the fraction: \(\lim _{x\rightarrow +\infty}\left(\frac{4+\frac{3}{x}}{\sqrt{2+\frac{3}{x}}}\right)\) - step3: Rewrite the expression: \(\frac{\lim _{x\rightarrow +\infty}\left(4+\frac{3}{x}\right)}{\lim _{x\rightarrow +\infty}\left(\sqrt{2+\frac{3}{x}}\right)}\) - step4: Calculate: \(\frac{4}{\lim _{x\rightarrow +\infty}\left(\sqrt{2+\frac{3}{x}}\right)}\) - step5: Calculate: \(\frac{4}{\sqrt{2}}\) - step6: Multiply by the Conjugate: \(\frac{4\sqrt{2}}{\sqrt{2}\times \sqrt{2}}\) - step7: Multiply the numbers: \(\frac{4\sqrt{2}}{2}\) - step8: Reduce the fraction: \(2\sqrt{2}\) La limitante de la expresión \( \frac{4x+3}{\sqrt{2x^{2}+3x}} \) cuando \( x \) se acerca a \( +\infty \) es \( 2\sqrt{2} \).