Question

Find the limits, if they exist, or type DNE for any which do not exist.

lim(x,y)→(0,0) 5x^2/(x^2+5y^2)
1) Along the x-axis:

2) Along the y-axis:

3) Along the line y=mx :

4) The limit is:

Answer 1

$(1)\\\lim_{(x, y) \rightarrow (0,0)}\left( \frac{5x^2}{x^2 + 5y^2}\right) \\ \textsf{Along the} \hspace{0.1cm} x\hspace{0.1cm} \textsf{axis}, y = 0\\ \begin{aligned} \lim_{(x, y) \rightarrow (0,0)}\left( \frac{5x^2}{x^2 + 5y^2} \right) &= \lim_{(x, y) \rightarrow (0,0)}\left( \frac{5x^2}{x^2 + 5(0)^2}\right) \\&= \lim_{(x, y) \rightarrow (0,0)} \left(\frac{5x^2}{x^2} \right)\\& = \frac{5}{1} = 5 \end{aligned}\\ (2)\\\textsf{Along the} \hspace{0.1cm} y\hspace{0.1cm} \textsf{axis}, x = 0\\ \begin{aligned} \lim_{(x, y) \rightarrow (0,0)}\left( \frac{5x^2}{x^2 + 5y^2} \right)&= \lim_{(x,y) \rightarrow (0,0)}\left( \frac{5(0)^2}{((0))^2 + 5y^2}\right) \\&= 0 \end{aligned}\\ (3)\\\textsf{Along the line} \hspace{0.1cm} y = mx\\ \begin{aligned} \lim_{(x, y) \rightarrow (0,0)} \left(\frac{5x^2}{x^2 + 5y^2}\right) &= \lim_{(x, y) \rightarrow (0,0)}\left( \frac{5x^2}{x^2 + 5m^2x^2}\right) \\&= \lim_{(x, y) \rightarrow (0,0)}\left( \frac{5}{1 + 5m^2}\right) \\& = \frac{5}{1 + 5m^2} \end{aligned}\\ (4) \\\textsf{Since the results in} \hspace{0.1cm}(1), (2), (3) \hspace{0.1cm}\textsf{are unequal}, \\\textsf{the limit does not exist.}$