Question

1. $limlimits_{x\to+infty}\frac{ln(1 + e^{x})}{5x}$

Explanation:

Step1: Simplify the expression

When \(x\to+\infty\), \(e^{x}\to+\infty\), so \(1 + e^{x}\sim e^{x}\). Then \(\lim_{x\to+\infty}\frac{\ln(1 + e^{x})}{5x}=\lim_{x\to+\infty}\frac{\ln(e^{x})}{5x}\) (since \(\ln(1 + e^{x})\) and \(\ln(e^{x})\) are equivalent as \(x\to+\infty\)).
Since \(\ln(e^{x})=x\), the limit becomes \(\lim_{x\to+\infty}\frac{x}{5x}\).

Step2: Calculate the limit

\(\lim_{x\to+\infty}\frac{x}{5x}=\lim_{x\to+\infty}\frac{1}{5}\) (cancel out \(x\) in the numerator and denominator).

Answer:

\(\frac{1}{5}\)