Y=xtanx/x^2+1 when is it discontinuous

Question

for me it is always continuous because theres the +1 in the denominator, however the back of my book says that the answer is all x except npi/2

Answer 1

x²+1 is never 0 if x is real.  However, tan x = sinx/cosx is undefined when x = ±π/2, ±3π/2, ...

 

So, the domain = {x l x ≠ ±π/2, ±3π/2, ...}

 

                      = {x l x ≠ nπ/2, where n is an odd integer}   

Answer 2

Hi, Pia,

 

You are correct in that the denominator x^2+1 doesn't become zero. However, tan(x)  can still be undefined. Remember that tan x = sin(x)/cos(x), so you need to determine when the cos x = 0. Recall from the unit circle that this would be when x = pi/2, 3pi/2, 5pi/2, etc. So the function would be discontinuous at those x values. I hope this helps. :)

 

Emmanuel