Question
Evaluate the limit using the substitution $z=x^{2}+y^{2}$ and observing that $z \rightarrow 0^{+}$ if and only if $(x, y) \rightarrow(0,0) .$$$\lim _{(x, y) \rightarrow(0,0)} \frac{1-\cos \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}$$
Step 1
Step 1: We are given the limit $$ \lim _{(x, y) \rightarrow(0,0)} \frac{1-\cos \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}} $$ We can substitute $z=x^{2}+y^{2}$, and observe that $z \rightarrow 0^{+}$ if and only if $(x, y) \rightarrow(0,0)$. Show more…
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