Limits of two variables
- Thread starter LuvMath
- Start date
I'm not so sure it'll do limits of two variables. I have a 92 and it doesn't.
As far as I know.
You must be able to interpret them.
For instance, let:
\(\displaystyle \L\\f(x,y)=\frac{xy}{x^{2}+y^{2}}\)
Find the limit of f(x,y) as (x,y)-->0 along
a: the x-axis
b:the parabola x^2
What do you do with your calculator?.
For a: the x-axis has parametric equations x=t and y=0, with (0,0) corresponding to t=0, so:
\(\displaystyle \L\\\lim_{\underbrace{(x,y)\to\(0,0)}_{\text{along y=0}}}f(x,y)=\lim_{t\to\0}f(t,0)=\lim_{t\to\0}\frac{0}
{t^{2}}=0\)
For part b:
The parabola \(\displaystyle \L\\y=x^{2}\) has parametrics \(\displaystyle x=t \;\ and \;\ y=t^{2}\), with (0,0) corresponding to t=0:
\(\displaystyle \L\\\lim_{\underbrace{(x,y)\to\(0,0)}_{\text{along y=0}}}f(x,y)=\lim_{t\to\0}f(t,t^{2})=\lim_{t\to\0}\frac{t^{3}}{t^{2}+t^{4}}=\lim_{t\to\0}\frac{t}{1+t^{2}}=0\)
See what I mean?. Perhaps there's a way to interpret it. I haven't delved into it.
As far as I know.
You must be able to interpret them.
For instance, let:
\(\displaystyle \L\\f(x,y)=\frac{xy}{x^{2}+y^{2}}\)
Find the limit of f(x,y) as (x,y)-->0 along
a: the x-axis
b:the parabola x^2
What do you do with your calculator?.
For a: the x-axis has parametric equations x=t and y=0, with (0,0) corresponding to t=0, so:
\(\displaystyle \L\\\lim_{\underbrace{(x,y)\to\(0,0)}_{\text{along y=0}}}f(x,y)=\lim_{t\to\0}f(t,0)=\lim_{t\to\0}\frac{0}
{t^{2}}=0\)
For part b:
The parabola \(\displaystyle \L\\y=x^{2}\) has parametrics \(\displaystyle x=t \;\ and \;\ y=t^{2}\), with (0,0) corresponding to t=0:
\(\displaystyle \L\\\lim_{\underbrace{(x,y)\to\(0,0)}_{\text{along y=0}}}f(x,y)=\lim_{t\to\0}f(t,t^{2})=\lim_{t\to\0}\frac{t^{3}}{t^{2}+t^{4}}=\lim_{t\to\0}\frac{t}{1+t^{2}}=0\)
See what I mean?. Perhaps there's a way to interpret it. I haven't delved into it.
