Answer:
To fill in the boxes, use these rules
In the second expression, y=x².
In the third expression, y=x
Step-by-step explanation:
Since y = 0 for all points on this path, then we have
[Tex]\lim_{(x,y) \to (0,0)} \frac{xy}{\sqrt{x^2+y^2}}=\lim_{x \to 0} \frac{x\cdot0}{\sqrt{x^2+0}}=0[/TeX]
Similarly, approaching along the y-axis yields a limit equal to 0. Since these two limits are the same, we will examine another approach path. Approach (0, 0) along the curve [tex]y = x^2[/TeX].
When x is positive, we have:
[Tex]\lim_{(x,y) \to (0,0)} \frac{xy}{\sqrt{x^2+y^2}}=\lim_{x \to 0} \frac{x^3}{\sqrt{x^2+x^4}}=\lim_{x \to 0} \frac{x^2}{\sqrt{x^2+x^2}}[/TeX]
From the equality above:
In the second expression, y=x².
In the third expression, y=x
When x is negative, we have,
[Tex]\lim_{(x,y) \to (0,0)} \frac{xy}{\sqrt{x^2+y^2}}=\lim_{x \to 0} \frac{x^3}{\sqrt{x^2+x^4}}=\lim_{x \to 0} \frac{x^2}{\sqrt{x^2+x^2}}[/TeX]
