Question

use the continuity of the exponential function to show that

lim x->0 (1 + tx)^(1/x) = e^t @Mathematics

Answer 1

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

Answer 2

Take the natural log of both sides, direct substitution, then solve for t.

Answer 3

what happens when I take the log of a limit?

Answer 4

why am I trying to solve for t?

Answer 5

Bahh it keeps bugging out!!!!

Answer 6

I don't even know how to begin going about solving this

Answer 7

\[\lim_{x \rightarrow 0} (1+tx)^{\frac{1}{x}}\]
Now let k=1/x
\[\lim_{k \rightarrow \infty } \left ( 1+\frac{t}{k}\right )^{k}\]
Now let k=ta
\[\lim_{a \rightarrow \infty } \left ( 1+\frac{1}{a}\right )^{at} = e^{t}\]