Limiting Euler

Today's question is about limits and Euler's number, e. This number has many interesting properties, and one of them is the derivative of e^x, which is e^x. Today's question does not deal with the derivative of this function, but with some limits.

Find the limit:
lim_{x -> infinity} (e^x + x)^1/x

This question is definitely not easy, mainly because of that annoying x in there and the 1/x exponent. However, we must never give up. We need to fight this question, so let's fight it with simplicity. Define another variable!

y = (e^x + x)^1/x

Simple, isn't it? All we have to do now is... well... find the limit of y. That didn't help. You can see there is a fraction in the exponent, so if we could just bring it down to a normal fraction, we could use L'Hopital's rule and solve this.

To get the exponent down, we can use a very well known technique: logs. There is a great rule in logarithms which states:
log_a bⁿ = nlog_a b

Using that fantastic rule we can bring the 1/x down and use L'Hopital's rule! To get a nice answer, let's take the logarithm of base e of this function, also written as ln (natural logarithm). So:
ln y = ln (e^x + x)^1/x

Using the logarithm rule mentioned earlier, we can say this equals:
ln (e^x + x) * 1/x
ln (e^x + x) / x

Mr. L'Hopital starts to smile. Now we can find the limit of this function and remember this is the logarithm of the function, so we'll need to consider that.

lim_{x -> infinity} ln y = lim_{x -> infinity} ln (e^x + x) / x

Using L'Hopital's rule (lim_{x -> a} f(x)/g(x) = lim_{x -> a} f'(x)/g'(x)), we can conclude that:
lim_{x -> infinity} ln (e^x + x) / x
is equal to:
lim_{x -> infinity} (e^x + 1) * (1 / (e^x + x)) / 1
lim_{x -> infinity} (e^x + 1) / (e^x + x)

Use L'Hopital's rule twice more:
lim_{x -> infinity} e^x / (e^x + 1)
lim_{x -> infinity} e^x / e^x

Now that looks familiar, doesn't it? This limit equals 1.

Remember: lim_{x -> infinity} ln y = 1, so lim_{x -> infinity} y = e

The limit of the entire function is e. Problem solved.

Hope you enjoyed,
Nadav

nadavs