Single Variable Calculus, Chapter 7, 7.8, Section 7.8, Problem 60
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Determine the $\displaystyle \lim_{x \to \infty} (e^x + x)^{\frac{1}{x}}$. Use L'Hospital's Rule where appropriate. Use some Elementary method if posible. If L'Hospitals Rule doesn't apply. Explain why.
If we let $y = (e^x + x)^{\frac{1}{x}}$, then
$ \ln y = \left( \frac{1}{x} \right) \ln (e^x + x)$
So,
$\displaystyle \lim_{x \to \infty} \ln y = \lim_{x \to \infty} \frac{\ln(e^x+x)}{x}$
By applying L'Hospital's Rule...
$\displaystyle \lim_{x \to \infty} \frac{\ln(e^x+x)}{x} = \lim_{x \to \infty} \frac{\frac{e^x+1}{e^x + x}}{1} = \lim_{x \to \infty} \frac{e^x +1}{e^x + x}$
If we evaluate the limit, we will still get indeterminate form, so we next apply L'Hospital's Rule once more. Thus,
$\displaystyle \lim_{x \to \infty} \frac{e^x +1}{e^x +x} = \lim_{x \to \infty} \frac{e^x}{e^x +1}$
Again, by applying L'Hospital's Rule...
$\displaystyle \lim_{x \to \infty} \frac{e^x}{e^x + 1} = \lim_{x \to \infty} \frac{e^x}{e^x} = 1$
Hence,
$\displaystyle \lim_{x \to \infty} \ln y = \lim_{x \to \infty} \frac{\ln(e^x+x)}{x} = 1$
Therefore, we have
$\displaystyle \lim_{x \to \infty} (e^x + x)^{\frac{1}{x}} = \lim_{x \to \infty} e^{\ln y} = e^1 = e$
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