Question

use the continuity of the exponential function to show that
lim x->0 of (1+tx)^(1/x) = e^t use the continuity of the exponential function to show that
lim x->0 of (1+tx)^(1/x) = e^t @Mathematics

Answer 1

\[\lim_{x \rightarrow 0}(1+tx)^{1/x} = e^t\]

Answer 2

I don't know how to use the continuity of the exponential function to show this

Answer 3

i guess i don't either. the limit is easy enough, but not sure what the question wants

Answer 4

how can I show that the limit equals e^t?
can I do it without using the definition of e?

Answer 5

without the definition of e it doesn't even make sense.
i would say
\[(1+xt)^{\frac{1}{x}}=e^{\frac{1}{x}\ln(1+xt)}\] and then take the limit in the exponent. maybe that is what the question is asking for, because that does in fact use "the continuity of the exponential function"

Answer 6

show that
\[\lim_{x\rightarrow 0}\frac{1}{x}\ln(1+xt)=t\] and you will have your solution

Answer 7

that's actually perfect! exactly what I was looking for, thank you

Answer 8

oh good.
yw

Answer 9

and solve via l'hopital i guess