Asked by Sean
Given that lim x-> +infinity of (1 + 1/x)^x = e
Show that lim x-> +infinity of (1 + k/x)^x = e^k
Show that lim x-> +infinity of (1 + k/x)^x = e^k
Answers
Answered by
Count Iblis
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
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
Answered by
Sean
Makes sense. Thanks Count!
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