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.

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