# How do you find the limit of #(arctan(x)) / (5x)# as x approaches 0?

##### 2 Answers

#### Explanation:

To find this limit, notice that both the numerator and denominator go to

By applying L'Hospital's rule, we take the derivative of the numerator and denominator, giving us

We can also check this by graphing the function, to get an idea what

Graph of

graph{(arctan x)/(5x) [-0.4536, 0.482, -0.0653, 0.4025]}

A lengthier approach using trig is explained below.

#### Explanation:

Just in case you're not comfortable with L'Hopital's Rule, or have not yet been exposed to it, another approach to solving the problem involves using the definition of the arctangent function.

Recall that if

From the diagram, it is clear that

Using this plus the fact that

This is equivalent to:

We know that