Question #2121

I need to find the limit of a problem and I keep getting zero as the answer with different paths. The problem is the limit (x,y) approaches (0,0) of (6x^3)/(2x^4 + y^4).

Expert's answer

Let's find the limit along the path y = y1 + m(x-x1), where x1 = 0 and y1 = 0.

<img src="/cgi-bin/mimetex.cgi?%5Clim_%7Bx%20%5Cto%200%7D%5Cfrac%7B6x%5E3%7D%7B2x%5E4%20+%20m%5E4x%5E4%7D%20=%20%5Cinfty" title="\lim_{x \to 0}\frac{6x^3}{2x^4 + m^4x^4} = \infty">

So, the limit equals to infinity.

<img src="/cgi-bin/mimetex.cgi?%5Clim_%7Bx%20%5Cto%200%7D%5Cfrac%7B6x%5E3%7D%7B2x%5E4%20+%20m%5E4x%5E4%7D%20=%20%5Cinfty" title="\lim_{x \to 0}\frac{6x^3}{2x^4 + m^4x^4} = \infty">

So, the limit equals to infinity.

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