Question #4ca29

1 Answer
Eddie
Feb 15, 2017

#= - 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.

Related questions
See all questions in Indeterminate Forms and de L'hospital's Rule
Impact of this question
1391 views around the world
You can reuse this answer
Creative Commons License