Question

In Exercises \(19-24,\) find the limit (if it exists). If the limit does not exist, explain why. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x+y}{x^{2}+y}\)

Short answer

The limit does not exist because the results are not consistent when approached from different paths.

Step-by-step solution

Approach along the y-axis

First, let's see what happens when we approach the point (0,0) along the y-axis. To do this, we set \(x=0\) in the function and look at \(\lim_{y \rightarrow 0} \frac{0+y }{ 0^2 + y}\), simplifying to \(\lim_{y \rightarrow 0} \frac{y}{y}= \lim_{y \rightarrow 0} 1 =1\).

Approach along the x-axis

Next, let's see what happens when we approach the point (0,0) along the x-axis. To do this, we set \(y=0\) in the function and look at \(\lim_{x \rightarrow 0} \frac{x + 0 }{ x^2 +0 }\), simplifying to \(\lim_{x \rightarrow 0} \frac{x}{x^2} = \infty\). Since the limit is not the same from these two paths, the limit does not exist.