Question

What is the limit as x approaches 0 of `tan(5x)/(4x)`?

Answer 1

If you try to insert the value in the function, you get `(0/0)`. This is a sign that you can use L'Hôpital's Rule. You can take the derivative of the nominator and the denominator and take the limit of that. It will be the same. Let's try it!

`lim_{x to 0} tan(5x)/(4x)`
First we'll derive the nominator:
`d/dx tan(5x) = sec^2(5x)*5` (chain rule)
Then the denominator:
`d/dx 4x = 4`

Now take the both of them together.
`lim_{x to 0} (sec^2(5x)*5)/4`
The definition of secand is: the reciprocal of cosine:
`lim_{x to 0} (5)/(cos^2(5x)*4)`
And now we can just plug in the value:
`5/(cos^2(5*0)*4)5/(cos^2(0)*4) = 5/(1*4) = 5/4`

Hope this helps.