Q.Evaluate $\displaystyle\lim_{x\to\infty}\left(1+\dfrac{2}{x}\right)^{x}$.

This resembles the standard limit defining $e$, $\lim_{x\to\infty}(1+\frac1x)^x=e$, but with $\frac2x$ in place of $\frac1x$ — a substitution converts it to the standard form.

Step 1 — Substitute to match the standard form. The standard limit is $\lim_{m\to\infty}\left(1+\dfrac1m\right)^m=e$. Here we have $\dfrac2x$ in place of $\dfrac1m$, so let $m=\dfrac{x}{2}$ (equivalently $x=2m$); then $\dfrac{2}{x}=\dfrac{1}{m}$, and as $x\to\infty$, $m\to\infty$ as well.

Step 2 — Rewrite the expression.

Step 3 — Apply the standard limit. As $m\to\infty$, $\left(1+\dfrac1m\right)^m \to e$, so the whole expression tends to $e^2$. …

The most common error is to directly write the answer as $e$ without noticing the $2$ in $\frac2x$ — the exponent that survives after the substitution (here, $2$) becomes the power to which $e$ is raised i …