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A scientist studies the expression F(x) = (1 + 3/x)²ˣ for large x…

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A scientist studies the expression $F(x) = \left(1 + \dfrac{3}{x}\right)^{2x}$ for large $x$ . As $x$ grows, the base tends to $1$ and the exponent tends to infinity. Find the value of $F(x)$ at $x = 1000$ , to two decimal places. Then find the limit of $F(x)$ as $x$ tends to infinity, to two decimal places. Write the exponent as $2x = 2 \cdot x$ and use the limit $\left(1 + \dfrac{3}{x}\right)^x \to e^3$ .

Need a reminder? The key ideaShow

$\displaystyle\lim_{x \to \infty}\left(1 + \dfrac{k}{x}\right)^x = e^k$ . If $f \to 1$ and $g \to \infty$ , then $f^g \to e^{\lim g(f - 1)}$ . Remove $\infty - \infty$ by a common denominator or by a conjugate.

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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