Limits — Question 1

Question 1 – Solution

Strategy. A polynomial in two variables is continuous everywhere, so its limit equals its value at the target point.

Step 1: Continuity Each monomial and constant is continuous, and finite sums of continuous functions are continuous. Direct substitution is therefore valid.

Step 2: Substitute $$\begin{align*} x^2y+3xy^2-2y+5 &=(2)^2(-1)+3(2)(-1)^2-2(-1)+5\\ &=-4+6+2+5=9. \end{align*}$$ Thus $$\boxed{\displaystyle \lim_{(x,y)\to(2,-1)}(x^2y+3xy^2-2y+5)=9}.$$

Verification. The four contributions are $-4$, $6$, $2$, and $5$; their sum is $9$. No denominator or restricted operation can fail at $(2,-1)$.