The limit as x approaches infinity of (1+1/x)^x.

I tried this twice and I got an answer of e both times, but the answer that my teacher read off in class was not e. Please help.

Lim (1 + 1/x)^x , as x approaches infinity is a definition of e

perhaps your teacher gave you a decimal approximation of e.

If not, then your teacher is wrong.

That makes me feel so much better. It must have been an error on the teacher's part then. Thank you very much.

To evaluate the limit as x approaches infinity of (1+1/x)^x, you can use the concept of the exponential definition of the number e.

The number e is defined as the limit of (1 + 1/n)^n as n approaches infinity. In other words:
e = lim(n->∞) (1 + 1/n)^n

Given that your expression is (1 + 1/x)^x, we can rewrite it as:
lim(x->∞) (1 + 1/x)^x
= lim(x->∞) [(1 + 1/x)^(x/x)]^x
= lim(x->∞) [(1 + 1/x)^x/x]^x
= [lim(x->∞) (1 + 1/x)^x/x]^x

Now, let's focus on the inner expression (1 + 1/x)^x/x as x approaches infinity. As x approaches infinity, the term 1/x approaches 0. So we rewrite the expression as follows:
lim(x->∞) [(1 + 1/x)^x/x]^x
= [lim(x->∞) (1 + 1/x)^(x/x)]^x
= [lim(x->∞) (1 + 0)]^x
= [1]^x
= 1

Therefore, the limit as x approaches infinity of (1+1/x)^x is 1. It is not equal to e. It seems that your previous result of e might have been a coincidence or a mistake, as the correct value is indeed 1.