Questions: lim(x -> ∞)(e^x + x)^(4/x)

Solution Steps

To solve the limit $\lim _{x \rightarrow \infty}\left(e^{x}+x\right)^{4 / x}$, we can use the properties of logarithms and limits. First, take the natural logarithm of the expression to simplify it. Then, evaluate the limit of the logarithm and exponentiate the result to get the final answer.

To simplify the limit, we take the natural logarithm of the expression: \[ \ln\left(\left(e^x + x\right)^{\frac{4}{x}}\right) = \frac{4}{x} \ln\left(e^x + x\right) \]

Next, we evaluate the limit of the logarithm as \( x \) approaches infinity: \[ \lim_{x \to \infty} \frac{4}{x} \ln\left(e^x + x\right) \] Since \( e^x \) grows much faster than \( x \), we can approximate \( \ln(e^x + x) \) by \( \ln(e^x) \): \[ \ln(e^x + x) \approx \ln(e^x) = x \] Thus, the limit becomes: \[ \lim_{x \to \infty} \frac{4}{x} \cdot x = 4 \]

Since we took the natural logarithm initially, we need to exponentiate the result to get the final answer: \[ \lim_{x \to \infty} \left(e^x + x\right)^{\frac{4}{x}} = e^4 \]

Final Answer