Lim [(1+x)^(1/x)] - e

x->0 x

Lim x -->0 [(1+x)^(1/x)] = e

To find the limit of the given expression as x approaches 0, we can use some algebraic manipulations and properties of limits.

First, let's simplify the expression [(1+x)^(1/x)] - e:

lim (x->0) [(1+x)^(1/x)] - e

Since we have a subtraction, let's split the expression into two separate limits:

lim (x->0) (1+x)^(1/x) - lim (x->0) e

Now, let's focus on evaluating each limit separately.

1. Limit of (1+x)^(1/x) as x approaches 0:

lim (x->0) (1+x)^(1/x)

This expression is of the form "1^∞", which is an indeterminate form. To evaluate this limit, we can take the natural logarithm of both sides:

ln [lim (x->0) (1+x)^(1/x)] = ln [lim (x->0) (1+x)^[(1/x)*(x)]]

ln [lim (x->0) (1+x)^(1/x)] = ln [lim (x->0) (1+x)^1]

ln [lim (x->0) (1+x)^(1/x)] = ln [lim (x->0) (1+x)]

Now, we can evaluate the limit on the right-hand side:

ln [lim (x->0) (1+x)] = ln (1+0) = ln 1 = 0

So, the limit of (1+x)^(1/x) as x approaches 0 is 1.

2. Limit of e as x approaches 0:

lim (x->0) e = e

Now, let's substitute these results back into our original expression:

lim (x->0) [(1+x)^(1/x)] - e = 1 - e

Therefore, the limit of [(1+x)^(1/x)] - e as x approaches 0 is 1 - e.