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$$ \lim_{x \to \infty}(1+4/x)^\sqrt{x^2+1} $$ is like
$$ \lim_{x \to \infty}(1+1/x)^x = e $$

I have replaced $\sqrt{x^2+1}$ by $x$ but I haven't got the expected result ($e^4$).

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    After replacing $\sqrt{x^2 + 1}$ by $x$, try raising to the power $\frac{1}{4} \cdot 4 = 1$ to get $$\lim_{x\to\infty} (1 + 4/x)^{x} = \lim_{x\to\infty} \left((1 + 4/x)^{x/4}\right)^{4} = \left(\lim_{x\to\infty} (1 + 4/x)^{x/4}\right)^4.$$ Then replace $x/4$ by $y$ to use your standard limit.2012-05-15

3 Answers 3