Question

How do you find the limit of `(sin (4x)) / (tan(5x))` as x approaches 0?

Answer 1

Use Algebra, trigonometry and the fundamental trigonometric limit.

Explanation:

`lim_(theta rarr0) sintheta/theta = 1`

We can use this to find `lim_(xrarr0) sin(7x)/x` because

`lim_(xrarr0) sin(7x)/x = lim_(xrarr0) 7/1sin(7x)/(7x)`

`= 7lim_(xrarr0) sin(7x)/(7x)`

Now, with `7x = theta` we see that `lim_(xrarr0) sin(7x)/(7x) = 1`

So we finish with

`7lim_(xrarr0) sin(7x)/(7x) = 7(1) = 7`#

We'll also need `lim_(theta rarr0) costheta = 1` (Cosine is continuous at `0`.)

`tan theta = sintheta/costheta`

Here is the solution

`sin(4x)/tan(5x) = sin(4x)/(sin(5x)/cos(5x))`

`= sin(4x)/1 cos(5x) 1/sin(5x)`

`= sin(4x)/4x * 4/1 cos(5x) 5x/sin(5x) * 1/5`

`= 4/5 [sin(4x)/4x] [cos(5x)] [5x/sin(5x)]`

Taking limits as `xrarr0` we get,

`= 4/5 [1][1][1] = 4/5`