Approximating end behavior of a function by plugging in infinity

Question

In Algebra 2, I learned to be able to tell if the end behavior of a function has an asymptote, approaches infinity, approaches zero, etc, by plugging in numbers closer and closer to infinity, or by plugging in infinity itself and checking the answer. For example, by plugging in infinity in the function $f(x) = \frac{1}{x}$, y would equal $\frac{1}{\infty} = 0$. So you know that the function approaches zero. For the function: $f(x) = (1+x)^{\frac{1}{x}}$, plugging in infinity for x gives an exponent that approaches zero. Since anything to the zero power equals one, I would expect the graph to approach one, however, it approaches $e$. What is the reason for this?

Answer 1

The reason, my friend is indeterminacy.

As you try to put $x=\infty$ in your function, you have something like $$(\rightarrow \infty)^{\rightarrow 0} $$ Which is an indeterminate form.

See, $\infty$ is not treated the way we treat numbers, and you can't say that $\infty^0=1$, However, we can apply the concept of limits, and see the behaviour of our function as $x$ approaches $\infty$, as you rightly stated, it goes to $e$.