Evaluating limits of several variables

[Odyssey]

Hi, in evaluating limits of several variables, is there a general method in approaching it? The plugging in the values method is easy, but the harder limits such as those 0/0 form...is there a general guideline to solving those problems?

How do I evaulate the following limits? (need tips and hints, not answer :tongue2:)
\lim_{\substack{x\rightarrow 0\\y\rightarrow 0}} f(x,y) = \frac{(x-1)^2\ln{x}}{(x-1)^2y^2}

\lim_{\substack{x\rightarrow 0\\y\rightarrow 0}} f(x,y)=\frac{x^2y}{x^2+y^2}

\lim_{\substack{x\rightarrow 0\\y\rightarrow 0}} f(x,y)=\frac{x^2+y^2-z^2}{x^2+y^2+z^2}

\lim_{\substack{x\rightarrow 0\\y\rightarrow 0}} f(x,y)=\frac{4xy}{3y^2-x^2}

Thank you for the help.

[matt grime]

Firstly, an easy check you must do is to see if the limit does genuinely exist.

Often, it doesn't for easy reasons, or you find the what the limit out to be in the checking.

To do this a standard technique is to let x and y tend to zero along some particular path, eg let x=y and tend to zero and then x=2y and let that tend to zero and see fi you get the same answer.