# A Beautiful Limits Problem

## A E-njoyable Problem

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This limit may look simple at first glance, but to solve it you will need to spot one beautiful step. I believe that doing maths is the only way to truly understand it, so I really suggest trying this problem before looking at the solution development below.

# Solution

This limit requires two main steps, first, we perform a special function, then use a standard limit evaluating method.

Firstly, we shall set this limit equal to a variable, let’s say J:

Now we are going to take the natural logarithm, which then will allow us to simplify this further.

Next, since the natural logarithm is a continuous function and defined for all numbers except zero, so as long as this is not equal to zero, we can slip it onto the inside giving:

Then we can use some laws of logarithms to simplify this further into:

Now we need to do the second step and use something called L’Hôpital’s rule, which states that if the limit has an indeterminate form that looks like zero over zero or infinity over infinity, we can take the derivative of the top and bottom. In this example, the top and bottom are both zero when x is zero, so this limit evaluates to:

Then we can just plug in that x is zero giving:

Which can be easily rearranged to give:

Which is the solution to our limit.

# Conclusion

I believe that everyone should be able to enjoy maths, and getting started solving problems is a great way to do this. So, if you did this limit using a different method, comment on how you did it. I would love to know and as always,

Have fun and never stop puzzling.

Also, why not look at my collection of puzzles below ↓↓↓

## Problems & Solutions

7 stories

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I have three Passions : Maths, Maths, Maths. Are these yours?