limit of sin^-1(xy-2)/tan^-1(3xy-6) as (x y) approaches to (2 1)

Question : Find the limit of sin^-1(xy-2)/tan^-1(3xy-6) as (x y) approaches to (2 1) \[{\rm limit}_{\left(x,y\right)\rightarrow(2,1)}{\frac{\sin^{-1}{\left(xy-2\right)}}{\tan^{-1}{\left(3xy-6\right)}}} \] Solution : We shall let the limit is \[ L={\rm limit}_{\left(x,y\right)\rightarrow(2,1)}{\frac{\sin^{-1}{\left(xy-2\right)}}{\tan^{-1}{\left(3xy-6\right)}}}\] Multiplying and dividing by xy-2 \[L={\rm limit}_{\left(x,y\right)\rightarrow(2,1)}{\frac{\sin^{-1}{\left(xy-2\right)}}{xy-2}\frac{\left(xy-2\right)}{\tan^{-1}{\left(3xy-6\right)}}}\] \[={\rm limit}_{\left(x,y\right)\rightarrow(2,1)}{\frac{\sin^{-1}{\left(xy-2\right)}}{xy-2}}{\rm limit}_{\left(x,y\right)\rightarrow(2,1)}\frac{\left(xy-2\right)}{\tan^{-1}{\left(3xy-6\right)}}\] Now multiplying…

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