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 )
\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.

 

[wais]

Hi there,

Yes, there is a general method for evaluating limits of several variables. It involves using the properties of limits, such as the limit laws, and applying them in a step-by-step manner. Here are some tips and hints for solving the limits you mentioned:

1. For the limit \lim_{\substack{x\rightarrow 0\\y\rightarrow 0}} f(x,y) = \frac{(x-1)^2\ln{x}}{(x-1)^2y^2}, you can start by factoring out (x-1)^2 in the numerator and denominator. Then, you can use the limit laws to evaluate the limit.

2. For the limit \lim_{\substack{x\rightarrow 0\\y\rightarrow 0}} f(x,y)=\frac{x^2y}{x^2+y^2}, you can use the Squeeze Theorem to show that the limit is equal to 0. You can do this by showing that the limit is bounded between 0 and a function that approaches 0 as (x,y) approaches (0,0).

3. For the limit \lim_{\substack{x\rightarrow 0\\y\rightarrow 0}} f(x,y)=\frac{x^2+y^2-z^2}{x^2+y^2+z^2}, you can use the fact that (x^2+y^2-z^2) is a difference of squares, and then factor it accordingly. This will allow you to simplify the expression and evaluate the limit.

4. For the limit \lim_{\substack{x\rightarrow 0\\y\rightarrow 0}} f(x,y)=\frac{4xy}{3y^2-x^2}, you can use the fact that (3y^2-x^2) is a difference of squares, and then factor it accordingly. This will allow you to simplify the expression and evaluate the limit.

Overall, when evaluating limits of several variables, it is important to use the limit laws and the properties of limits to simplify the expressions and make them easier to evaluate. Remember to always check for any indeterminate forms (such as 0/0) and use appropriate techniques to handle them. I hope this helps!