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I encountered a question where I have to find the limit of $$ a)\,\,[1+(1/x)]^{x^2/(x+y)} \text{ as }x\to\infty\text{ and }y\to 0\\ b)\,\,[1+(y/x)]^x\text{ as }x\to\infty\text{ and }y\to k $$ These two functions bring in my mind the formula : As $x \to \infty$, $(1+1/x)^x\to e$ But I don't know how to use it .

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In editing your post, I fixed your claim that $x\to\infty$ implies $1+1/x\to e$. – Ian Coley Mar 18 '13 at 22:33

One idea is to make the unusual object look as much like the familiar object as possible. For example, in the first case, rewrite as $$\left(1+\frac{1}{x}\right)^{x/(1+y/x)},$$ and then as $$\left(\left(1+\frac{1}{x}\right)^{x}\right)^{1/(1+y/x)}.$$ Note that we don't even need for $y$ to go to $0$, as long as it is bounded. The strategy for the second is very similar. Let's assume $k$ is positive. Rewrite our expression as $$\left(\left(1+\frac{y}{x}\right)^{x/y}\right)^{y},$$ and let $t=y/x$.

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