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This isn't a coursework problem. I'm on winter break.

A common approximation used in physics is:

(1+x)

This implies that

lim(x→0) (1+x)

which is a true statement. However,

lim(x→0) (1+x)

= lim(x→0) [(1+x)

= lim(x→0) e

This leads to the fact that

(1+x)

The question is: which one is a better approximation, and how do I show it?

Compute the error terms.

E

E

They intersect at E

1+nx = e

I'm not sure how to go from here. I'm pretty sure that x can't be isolated. However, since e

http://www.wolframalpha.com/input/?i=1%2Bnx+%3D+e^%28nx%29

Assuming positive n, for a large x, e

Assuming negative n, for a large x, e

Or it may just be the case that 1+nx is always a better approximation, since it's more commonly used.

## Homework Statement

A common approximation used in physics is:

(1+x)

^{n}≈ 1+nx for small xThis implies that

lim(x→0) (1+x)

^{n}= lim(x→0) 1+nxwhich is a true statement. However,

lim(x→0) (1+x)

^{n}= lim(x→0) [(1+x)

^{1/x}]^{xn}= lim(x→0) e

^{xn}This leads to the fact that

(1+x)

^{n}≈ e^{xn}for small xThe question is: which one is a better approximation, and how do I show it?

## The Attempt at a Solution

Compute the error terms.

E

_{1}= (1+x)^{n}- (1+nx)E

_{2}= (1+x)^{n}- e^{xn}They intersect at E

_{1}= E_{2}, or at1+nx = e

^{xn}I'm not sure how to go from here. I'm pretty sure that x can't be isolated. However, since e

^{xn}is always positive, I think the only root is at x=0. Wolfram gives a funny answer though. If anyone would like to explain Wolfram's answer, please do.http://www.wolframalpha.com/input/?i=1%2Bnx+%3D+e^%28nx%29

Assuming positive n, for a large x, e

^{xn}is a worse approximation for the function. Since e^{xn}grows much faster than (1+nx), e^{xn}is always a worse approximation than (1+nx).Assuming negative n, for a large x, e

^{xn}is a worse approximation for the function. Since e^{xn}doesn't grow as quickly as (1+nx) (it levels off), e^{xn}is always a better approximation than (1+nx).Or it may just be the case that 1+nx is always a better approximation, since it's more commonly used.

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