Question

Given that lim x-> +infinity of (1 + 1/x)^x = e
Show that lim x-> +infinity of (1 + k/x)^x = e^k

Answer 1

If:

Lim x --> a of f(x) = L

and g(x) is continuous at x = L, then:

Lim x --> a of g[f(x)] = g(L)

In this case we can take

f(x) = (1+1/x)^x

and

g(x) = x^k

We then have:

Lim x to +infinity of (1+1/x)^(k x) =e^k

We can write

Lim x to +infinity of (1+1/x)^(k x) =

Lim x to +infinity of (1+k/y)^(y)

by substituting y = k x

Answer 2

Makes sense. Thanks Count!