Question

find the limit (if it exists). (if an answer does not exist, enter dne.)
\\(\lim_{(x,y)\to(0,0)}\frac{x + y}{x^{8}+y}\\)

Explanation:

Step1: Approach along the line y = 0

Substitute y = 0 into the function.
\[

$$\begin{align*} \lim_{(x,y)\to(0,0)}\frac{x + y}{x^{8}+y}&=\lim_{x\to0}\frac{x+0}{x^{8}+0}=\lim_{x\to0}\frac{x}{x^{8}}=\lim_{x\to0}\frac{1}{x^{7}} \end{align*}$$

\]
As \(x\to0\), \(\frac{1}{x^{7}}\) does not exist (it approaches \(\infty\) or \(-\infty\) depending on the sign of \(x\)).

Step2: Conclusion

Since the limit is not the same along different paths (here we just showed non - existence along one path), the limit of the function as \((x,y)\to(0,0)\) does not exist.

Answer:

DNE