# Does this limit exist lim e^(r^2)/(cos(t)sin(t)) for (r,t)->(0,0)

1. Apr 12, 2005

### stunner5000pt

In my exam the question was to determine the existence of this limit
$$\lim_{(r,t)\rightarrow(0,0)} \frac{e^{r^2}}{\cos{t}\sin{t}}$$

now i wrote the numerator has no t associated with it so grows or shrinks without bounds, the same applies for the denominator...
so the limit does not exist

is this a good reason

another question was
$$\lim_{(x,y)\rightarrow(0,0)} \frac{\sin{xy}}{xy}$$
i substituted xy = u and got $$\lim_{u \rightarrow 0} \frac{\sin{u}}{u} = 1$$

is this the correct method?
Am i right?

2. Apr 12, 2005

### HallsofIvy

Staff Emeritus
The numerator does not "grow or shrink without bound". As r goes to 0, the numerator goes to 1.

However, you are correct that the denominator goes to 0 as t does no matter what r is. Since the numerator goes to 1, what does that tell you about the fraction?

I was a bit suspicious about your second method, but yes, it works, since u is a variable, you are not assuming any relationship between x and y.