peicewise limit

[markraz]

Hi, this limit problem x->pi/4.... which piecewise function do I use to solve the limit and why?
I would think I should use the Tan(x) but the answer to this problem is D which tells me
it should be the sin(45) or cos(45) which is sqr(2)/2...
I'm so confused. what is the basic procedure for 2sided piecewise limits such as this once?

the left hand and right hand limits would each give me answer sqr(2)/2 which would mean
the two sided limit would automatically be the same? if so what even is the point of the tan function here?
in what scenario would the x=45° ever get used?

thanks in advance

 

[Quaid]

which piecewise function do I use to solve the limit and why?

I would think I should use the Tan(x) but the answer to this problem is … sqr(2)/2

what is the basic procedure for 2sided piecewise limits

the left hand and right hand limits would each give me answer sqr(2)/2 which would mean the two sided limit would automatically be the same?

what is the point of the tan function here

Hi Mark:

With any limit where x approaches a constant value, you need to consider what's happening from either side. So, you treat the limit in this exercise like any other; the piecewise definition does not change the approach.

Yes, you are correct. Function f approaches the same value (√2/2), as x approaches Pi/4 from the left and from the right. When a function approaches the same fixed value from each side, then that value is the limit.

There is no special point to defining the function in three pieces; that's just the way it is.

Cheers :cool:

 

[Quaid]

in what scenario would the x=45° ever get used?

Well, anything having to do with f(Pi/4) would require you to use that piece.

Function f is an example of a "point-discontinuity", so an exercise about continuity would require you to consider what's up when x=Pi/4. :cool: