Can someone help with this limits question?

[losfomot]

Here is the question:

 

[math]\lim_{x\to\infty}\left(1+\frac{2}{x}\right)^x[/math]

 

 

I thought that (1 + 2/x) as x approaches infinity is just 1...

 

and 1 to the power of anything (including x) is still 1.

 

But it turns out that 1 is the wrong answer.

 

What's up with that?

[TD]

By defnition (or equivalent with possible other definitions), we have that

 

[math]\mathop {\lim }\limits_{x \to \infty } \left( {1 + \frac{1}

{x}} \right)^x = e[/math]

 

Furthermore, we also have that in general, with m a real number

 

[math]\mathop {\lim }\limits_{x \to \infty } \left( {1 + \frac{m}

{x}} \right)^x = e^m [/math]

 

So in your case, the answer would be [math]e^2 [/math].

 

To prove this, you can take the natural logarithm (ln/log) of the expression, the limit will then give the exponent of e.