limit of arcsin/arctan in two variables

May 2010
7
0
what is the steps for calculating lim (x,y)->(1,3) [arcsin(3xy-9)/arctan(xy-3)]
i know how to do it in 1 dimension..if let z=xy. can i actually do this?
my solution is 3

actually how to use \(\displaystyle code in this forum?
i am new here :)

thanks!\)
 

mr fantastic

MHF Hall of Fame
Dec 2007
16,948
6,768
Zeitgeist
what is the steps for calculating lim (x,y)->(1,3) [arcsin(3xy-9)/arctan(xy-3)]
i know how to do it in 1 dimension..if let z=xy. can i actually do this?
my solution is 3

actually how to use \(\displaystyle code in this forum?
i am new here :)

thanks!\)
\(\displaystyle
Yes. Yes. Click on the relavant link in my signature.\)
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Since x and y only appear as "xy", you can make that substitution. No matter how (x, y) approaches (1, 3), z= xy approaches 3.

If it had been something like "[arcsin(3xy-9)/arctan(y-3)]" you would not have been able to do that.
 
May 2010
7
0
Since x and y only appear as "xy", you can make that substitution. No matter how (x, y) approaches (1, 3), z= xy approaches 3.

If it had been something like "[arcsin(3xy-9)/arctan(y-3)]" you would not have been able to do that.
what should i write to express this?
just write let z=xy, when (x,y)->(1,3), z->3
then continue like one var?

if something like this, how can i calculate?
thanks~!
 
Jan 2010
354
173
You would just say something like

Let \(\displaystyle z=xy\). So if \(\displaystyle (x,y) \to (1,3)\), then \(\displaystyle z \to 3\).

Therefore:

\(\displaystyle \lim_{(x,y)\to(1,3)} \frac{\arcsin(3xy-9)}{\arctan(xy-3)} = \lim_{z \to 3} \frac{\arcsin(3z-9)}{\arctan(z-3)} = \cdots\)

You could continue solving the limit using standard one variable techniques.
 
  • Like
Reactions: HallsofIvy