Question

Question #4ca29

Answer 1

`= - 1/2`

Explanation:

It looks like:

`lim_(x to pi/4) (tan x - 1)/(sin 4x)`

We can use L'Hôpital's Rule because this is in indeterminate form, ie if we plug the x-value straight in, we get `(tan x - 1)/(sin 4x) = (1-1)/(0) = 0/0`.

So after one round of L'Hôpital, we are at:

`= lim_(x to pi/4) (sec^2 x)/(4 cos 4x)`

`= lim_(x to pi/4) (1)/(4 cos^2 x cos 4x)`

If we plug the x-value straight in again, we get:

`(1)/(4 cos^2 (pi/4) cos pi) = (1)/(4 (1/2) (-1)) = - 1/2`

Because `cos^2 x` and `cos 4x` are continuous at `x = pi/4`, we can rely on that result.

Of course, that's a complete black-box. The more transparent way is to use, perhaps, a Taylor Expansion; or, maybe some algebraic manipulation of the trig functions.