(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Evalute:

a) lim (x->0) (arctan x)/x

b) lim (x->1) (arctan(x) - pi/4)/(x-1)

2. Relevant equations

Inverse tangent, trig identities. Kline's calculus, which I am teaching myself from, does not have that much detail on limits.

3. The attempt at a solution

For (a), I solved it fairly easily. I used y = arctan(x), x = tan(y) to rewrite the limit as

y/tan(y) = y/(sin y/cos y) = y(cosy)/sin y) = (y/sin y)(cos y).

as x->0, y/sin y = 1, as does cos(y).

Applying the same approach to (b) has not rewarded me with a solution.

Making the same substition y = arctan(x), x=tan(y), I get

(y - pi/4)/(tan(y) - 1)

(4y - pi)(cos y)/(4(sin y - cos y))

But I don't seem to be able to find my way to a solution.

Some things I have tried:

trig identities like -cos(2a) = sin^2 y - cos^2 y, as well as others.

I tried dividing up the fraction, and computing the sums:

y/(sin y - cos y) - pi/(sin y - cos y).

I also tried finding the limit of the difference between this and part (a), on the assumption that the limit of the sums is the sum of the limits.

If anyone could prod this old guy in a useful direction, it would be appreciated.

Thanks,

Sheldon

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# Limit of arc tan.

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