Finding the limit for exponential function using Taylor Expansions

[georgetown13]

Homework Statement

Determine the limit and then prove your claim.

lim_{x$$\rightarrow$$$$\infty$$} (1+$$\frac{1}{x^2} }$$) ^x

Homework Equations

I know that the formal definition that I need to use to prove the limit is:

{lim_{x$$\rightarrow$$$$\infty$$} (1+$$\frac{1}{x^2}$$)^x=1}={$$\forall$$ $$\epsilon$$>0, $$\exists$$ N > 0, $$\ni$$ x>N $$\Rightarrow$$ |f(x)-1|< $$\epsilon$$}

The Attempt at a Solution

We have to use Taylor Expansions to find the Taylor polynomial of f(x) and bound the errors to solve for $$\delta$$, given $$\epsilon$$ >0.
The "x" exponent, however, is throwing me off. Could someone help guide me through the Taylor expansion of f(x)? I'd greatly appreciate it!

 

[LCKurtz]

Homework Statement

Determine the limit and then prove your claim.

lim_{x$$\rightarrow$$$$\infty$$} (1+$$\frac{1}{x^2} }$$) ^x

Homework Equations

I know that the formal definition that I need to use to prove the limit is:

{lim_{x$$\rightarrow$$$$\infty$$} (1+$$\frac{1}{x^2}$$)^x=1}={$$\forall$$ $$\epsilon$$>0, $$\exists$$ N > 0, $$\ni$$ x>N $$\Rightarrow$$ |f(x)-1|< $$\epsilon$$}

The Attempt at a Solution

We have to use Taylor Expansions to find the Taylor polynomial of f(x) and bound the errors to solve for $$\delta$$, given $$\epsilon$$ >0.
The "x" exponent, however, is throwing me off. Could someone help guide me through the Taylor expansion of f(x)? I'd greatly appreciate it!

Not offering to try the Taylor thing here; are you required to do it that way? Otherwise, I would let

$$y = \left( 1 + \frac 1 {x^2}\right)^x$$

and work with $\ln(y)$ using L'Hospital's rule.