Problem 14 - Calculus Early Transcendentals

Question

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(2,-1)}\left(x y^{8}-3 x^{2} y^{3}\right)$$

Step-by-Step Solution

Answer

Answer: The value of the limit is 14.

Step 1: Substituting the values of x and y

Substitute x=2 and y=-1 into the function. $$\lim _{(x, y) \rightarrow(2,-1)}\left(x y^{8}-3 x^{2} y^{3}\right)=\left(2(-1)^{8}-3(2)^{2}(-1)^{3}\right)$$

Step 2: Simplify the expression

Simplify the expression by calculating the powers and the multiplication. $$\left(2(-1)^{8}-3(2)^{2}(-1)^{3}\right)=(2(1)-3(4)(-1))$$

Step 3: Complete the calculation

Finish the calculation to find the limit. $$(2(1)-3(4)(-1))=(2+12)=14$$ Thus, the limit is 14: $$\lim_{(x, y) \rightarrow(2, -1)}\left(x y^{8}-3 x^{2} y^{3}\right) = 14$$