Answer to Question #288855 in Calculus for Pankaj

Question

lim x ➡0 (lim y ➡0 for ysinx/|x|) does not exist. Say true or false

Answer 1

$$\lim_{x \rightarrow 0 } \bigg(\lim_{y \rightarrow 0 } \frac{y \sin x}{|x|}\bigg)\\ \lim_{x \rightarrow 0 } \bigg(\lim_{y \rightarrow 0 } \frac{0 \cdot \sin x}{|x|}\bigg)\\ \lim_{x \rightarrow 0 } \bigg(\lim_{y \rightarrow 0 } \frac{0 }{|x|}\bigg)\\ \lim_{x \rightarrow 0 } \bigg(\lim_{y \rightarrow 0 } 0 \bigg)\\ \lim_{x \rightarrow 0 } \big(0 \big)\\ \lim_{x \rightarrow 0 } 0 =0\\$$

Therefore,

$$\lim_{x \rightarrow 0 } \bigg(\lim_{y \rightarrow 0 } \frac{y \sin x}{|x|}\bigg) =0\\$$

and thus exist (FALSE)