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.