help me to find limit

[Guest]

ı couldnt find result of the question can u guys help me

[Guest]

The first thing I would do is check to see if the limit exists. If the limit exists then the limit, as (x, y) goes to (0,0) along every possible path, must be the same. You cannot check "every possible path" but if checking just two gives different results, that proves that the limit does NOT exist.

If we take y= 0, [tex]\frac{sin(xy)+ xe^x- y}{x cos(y)+ sin(2y}[/tex] becomes [tex]\frac{sin(0)+ xe^x- 0}{x cos(0)+ sin(0)}= \frac{xe^x}{x}= e^x[/tex] which goes to 1. The limit as we approach (0,0) along the x-axis is 1.

If we take x= 0, [tex]\frac{sin(xy)+ xe^x- y}{x cos(y)+ sin(2y)}[/tex] becomes [tex]\frac{sin(0)+ 0e^0- y}{0 cos(y)+ sin(2y)}= \frac{-y}{sin(2y)}[/tex]. Now, I recall that [tex]\lim_{\theta\to 0}\frac{\theta}{sin(\theta)}= 1[/tex]. If we take [tex]\theta= 2y[/tex] then [tex]y= \frac{\theta}{2}[/tex] so that becomes [tex]-\frac{2y}{2sin(2y)}= -\frac{1}{2}\frac{2y}{sin(2y)}= -\frac{1}{2}\frac{\theta}{sin(\theta)}[/tex] and the limit, as (x,y) goes to (0,0) along the y-axis, is [tex]-\frac{1}{2}[/tex], not 1.

Perhaps you can't find the limit because there is no limit!