# limit of arcsin/arctan in two variables

#### pokemon1111

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
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 $$\displaystyle code in this forum? i am new here thanks!$$
$$\displaystyle Yes. Yes. Click on the relavant link in my signature.$$

#### drumist

• mr fantastic

#### pokemon1111

thanks all
why can i do so? i mean from 2 var->1 variable

#### HallsofIvy

MHF Helper
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.

#### pokemon1111

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~!

#### drumist

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.

• HallsofIvy

#### pokemon1111

thank you very much =)