Hello,
plz try to do these questions.
Have you attempted doing these limits? Take certain paths and determine whether the limits are the same or not; if not, then there is a contradiction and the limit D.N.E., and if it does then you can use approximating methods to determine whether it is a limit. Note that even if you find 3 paths with the same limit, this does not necessarily mean that the limit exists. You have to prove it. To show this, consider the 'squeeze principle':
|f(x,y) - L| ...
notice that with the first limit, we can rewrite it a bit using limit laws.
limit{x,y,z-->0,0,0} [e^(xyz) * sin(xyz)]/xyz
= limit{x,y,z-->0,0,0} e^(xyz) * sin(xyz)/xyz
= limit{x,y,z-->0,0,0}e^(xyz) * limit{x,y,z-->0,0,0} sin(xyz)/xyz by limit theorem for multiplication of limits
now, from calc 1, we know that limit{x-->0} sinx/x = 1
and e^0 = 1
so limit{x,y,z-->0,0,0}e^(xyz) * limit{x,y,z-->0,0,0} sin(xyz)/xyz = 1*1 = 1
and another thing. you have to look at the way the question is phrased. it can take a while to find a contradiction and show that the limit does not exist, so make sure you know that it doesn't exist before you do that. the question you had said:
"Find the limit of the following:"
that means, they know the limit exists, and they want you to find it. trying to find contradictions are a waste of time.
when do you know to look for contradicitions?
well, if the question said something like:
"do the following limits exists?"
or
"find the limit of each of the following, if they exist"
or something to that effect. if the question hints at uncertainty as to whether the limits exist, look for contradictions, if not, try proving the limit using the definition. but just plugging in numbers is usually the best way to start. and always look if the function can be split up into products or sums where the limit exist as we did in the first.